applications of method of least squares to the adjustment

Scholars' Mine Scholars' Mine

Professional Degree Theses Student Theses and Dissertations

1922

Applications of method of least squares to the adjustment of level Applications of method of least squares to the adjustment of level

circuits: Adjustment of elevations in Dawson and St. James circuits: Adjustment of elevations in Dawson and St. James

Townships, Phelps County, Missouri Townships, Phelps County, Missouri

Clarence Edward Bardsley

Follow this and additional works at: https://scholarsmine.mst.edu/professional_theses

Part of the Civil Engineering Commons

Department: Department:

Recommended Citation Recommended Citation Bardsley, Clarence Edward, "Applications of method of least squares to the adjustment of level circuits: Adjustment of elevations in Dawson and St. James Townships, Phelps County, Missouri" (1922). Professional Degree Theses. 165. https://scholarsmine.mst.edu/professional_theses/165

This Thesis - Open Access is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Professional Degree Theses by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected].

https://library.mst.edu/

https://library.mst.edu/

https://scholarsmine.mst.edu/

https://scholarsmine.mst.edu/professional_theses

https://scholarsmine.mst.edu/student-tds

https://scholarsmine.mst.edu/professional_theses?utm_source=scholarsmine.mst.edu%2Fprofessional_theses%2F165&utm_medium=PDF&utm_campaign=PDFCoverPages

http://network.bepress.com/hgg/discipline/252?utm_source=scholarsmine.mst.edu%2Fprofessional_theses%2F165&utm_medium=PDF&utm_campaign=PDFCoverPages

https://scholarsmine.mst.edu/professional_theses/165?utm_source=scholarsmine.mst.edu%2Fprofessional_theses%2F165&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

APPLICATIONS OF TIlE METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.

ADJUSTMENT OF ELEVATIONS IN DAWSON

AND ST. JAMES TOWNSHIPS, PHE['PS COUNTY,

MISSOURI.

BY

CLARENCE EDWARD SOLOMAN BARDSLEY

A

THE SIS submitted to the facul~y of the

SCHOOL OF MINES AND METALLURGY OF THE UNIVERSITY OF MISSOURI iB partial fulfillment of the work required for the

D E G R E E 0 F CIVIL ENGINEER

Rolla,Mo. 1922.

ASSISTEO BY

CHARLES EDWARn COOKE

~y-,-,r~ 1d,",-~ , PrG seor ef €ivil IRaiae.ria,.

APPLICATIONS OF THE METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.

ADJUSTMENT OF . ELEVATIONS IN DAWSON AND ST. J!MES TOWNSHIPS,

PHELPS COUNTY, MISSOURI~

CON TEN T S •

INTRODUCTION. Remarks on the Introduction. • • .. . . . · . . • • • • Liscrepancies among Observations •••••••••• • Carefulness of Observation • • • • • • • • • • • • • •

• • • · . . . . . Types and Occurance of Errors • • • • Circ.umstances affectin~ Observations Princip~e of Least Squares Stated • •

•. . • • • • • • • ix·

• • · . . . . . . .On the Uniformity of Least Square Results • • • • • • • Not in General Use Among Engineers • • • • • • • • • • Precision, Subsequent Measurements Dependent on results Theory "Stated for Weigbted Observations • • • • • • • • Acknowledgments • • • • • • • • • • • • • • • • • • • • x

PART. I. THE INDEPENDENT ADJUSTMENT OF A MULTIPLE LINE.

METHOD OF LEAST SQUARES APPLIED TO THE ADJUSTMENT O~ THE LINE A VIA B, C, AND D TO E.

BASED ON THE LINE OF LEV£LS RUN 8Y THE U. S. Q£otOGteAl SURVEY IN 1908 ALONG THE FRISCO R. R.

W IT K PER IlIA In lilTS Ellie H. ARK S , - A - A·Y ST. "A" E $, AND -E- AT KNOBYIEW •

. Multiple Lines of Levels •• • • • • • • • • • • • •• 2 sta.tement and Solution for .Adjust~ent Correction ••• 3 ~eneral Principle of Lea~t B~u8res • • • • • • • • • • 4 The Weights fe>r Level Lines .. • • • • • • • • • • • •• 4 Probable Error of Wei~hted Arithmetic Meall. • • • •• 4 Intermediate Points on Level Cireuits •••

IBterBiediate junction pointsB and, C • /

-i-

• • • • • •

• • • • • •

4

5

CON TEN T S •

PAR T I I . . TilE LEAST SQUARE ADJUSTMENT OF A LEVEL NET

BY MET BOD OF OBSERVATION EQUARIONS. APPLIED TO THE ADJUSTMENT

OF THE DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,

PHELPS COUNTY, MISSOURI.

lASED ON THE A VIA 8, C, AND 0 TO E BASE.

Observed Differences of Elevation by Links • . . . . . . 7 7 7 8

8

9

Level Base Lines • • • • • • • . . . . . . · .. . . . . . On the Solution of the Problem by Least Squares Statement of the Conditions . . . . . . . . . . . Observation Equations • • • • • 0 • • • • • • • •

Computation of Jnct.Elevations on Arranged Circuit 0 . . Rule: Formation of Normal Equations from

Observation Equations 0.. 10 Solution of Normal Equations

Elimination by Addition and Subtraction .. . . . Check on Residual Values • . . . . . . . . . . . • 0 0

Summary of Residual Values 0 0 0 • • • • • • • •

- -

• 12 o 15 • 15

Calculation and Tabulation of Adjusted Jnctopoints 0 •• 15 solution of Normal Equations by the P,q,r Method 0 0 0 0 16

(0) Normal Equations 0 0 • 0 •• 0 ••• 0 • 0 • 0 16 (P) Solution of Normal Equations 0 • • 0 • 0 • 0 • 16 (S) Most Probable Differences of Elevation • • • 0 16 (T) Most Probable Elevations • • • • 0 • • • • 0 • 16 (q) Solution of Normal Equations 0 • • • • • • 0 • 17 (r) Solution of Normal Equations • • • 0 0 0 ••• 17 Adjustment Quantities • 0 0 • 0 0 0 0 0 0 0 0 17 Check on Resid uals " • • 0 • • 0 • 0 • • 0 0 • 0 0 17

- -Adjustment of Intermediate Junction points o • o 0 • 18

Adjustment of Points Land K 0 00 0 • . • • . . • 18 - -Adjustment of point P •• • . 0 • 0 • • • e · • • • 19 Adjustment of Points Nand 0 . .. • • · . . . o • • 20

-ii-

• • • • • • • 21 • • • • • • • 21

C 0 K TEN TS • Adjustml:)nt of Spur Linl:)s PQ, NR, and OSO ; ••

Adjustml:)nt of Point Q • • • • • • • • • • Paralll:)l Duplicate Line of Levels • • • • • • • • • 21

Unweighted Measureml:)nts . . . . .. . . . . . • • • • • • • 22 probable Error of a Singll:) Ml:)asureml:)nt. • • • • • • • • 22 Probable Error of n Ml:)asurements •• • • • • • Adjustment of Point~ R •• • • • • • • • • • •

• • • • 22 • • • • 22

Adjustment of point.S • • • • • • • • • • • • • • • • 23 Duplicatl:) Line of Levels • • • • . . . . . ')3 · . . . '-_. -

Summary of A.II Junction point Eleavations . . . . • ••• '24 Adjustment of Intermediate Bench Marks • • . . • • • 25

Description of Benchmarks and their Adjusted Elevations Adjustment of Benchmarks Between "A" ..and "I ". -

" " " " " " " " I and P - -

It " II " " " " " p and Q

" " " " " " " " A and D

PAR T I I I • THE LK4..ST SQtl.~RE ADJUSTMENT OF' A LEVEl. NET

BY METHOD OF OBSERVATION EQUATIONS. APPLIED TO THE ADJUSTMENT



BASED ON THE LINE OF LEVEl8RUN BY THE

•

· •

• • • 26 • • · 29

· • • 31 • • • 34

U. S. GEOLOGICAL SURVEY IN 1908 ALONG THE FRI8CO R. R. WITH PERMANENT BENCHMARKS,-A- AT ST. JAMES,AND

-E- AT KNOBV lEW.

computation of Elevations in Arranged Circuit •••• Closures in Arranged Circuit .. . . .

Observation Equations •••••• • • • • •• • • • Rule: Formation of Normai Equations from

Observation Eq~atJons •• statl:)me~t on the Solution of Normal Equations .. • • •

• •

· . • •

• • • •

39 39 41

41 41

Formation of Normal Equations for Level Net. • • • • •• 42 Summary of Normal Equations showing Redundant Numbl:)rs.. 44

(P) solution of Normal Equations '. • • •• • • ••• 45 .( q) solution of Normal Equations. '.. • • • ,. • •• 46 (r) solution of Normal Equations • '. • • • • • • •• 47

- -Adjusted Elevations of Junction Points ,. • • • • •• •• 47

-iii-

CON TEN T S •

PAR T I V • THE LEAST SQHARE ADmSTMENT OF A LEVEL NET

flY METHOD OF CONIHTION EQUATIONS AND

CORRELATIVES. APPLIED TO THE ADJUSTMENT



BASED ON THE LINE OF LEVELS RUN BY THE U. S. GEOLOGICAL SURVEY IN 190B ALONa THE FRISCO R. R.

11TH PERMAN£NT IENCHMARKS -A- AT ST. JAMES, AND wE- AT KNDIVIEI.

Tabulation': ,Linka, Observed Diff • in Elevation, Distances, ,Weights, Deri ved Correction, Corrected Diffs. in

Condi tion Equations • • • • • • • • • • • • • • • • • • • On the formation of Condition Equations

Table of Correlat~s and, Corrections to Links • • • • • • •

Elev.50 •• 50 •• 50

. .. . . . . . . 51 Rules: Formation of Normal Equations from Condition Equations 51 Normal Equations. • .• • • • • • • • • • • • • • • • • • • •• 52 Notes on the (P),(q),(r) Solution of Normal Eq~ations •••• 52

(P)Sollltion of Normal Equations. • • • • • • • • • •• 52 (q) Solution of Normal Equations •• '. • • • • • • • •• 53 (r) or Back solution of Normal Equations. • • • • • •• 53

- -Adjusted Elevations of Junction points •••••• , •••••• Abridged Method A for the solution of Normal Equations . ' •••

Normal Equations with Constant and .check 'Columns .• • • • Abridged Solution A with Notes on the precess •••••• Correlat~s with Notes on the Pcocess of forming Them •• Discussion: Substitute for the (P)' (q), (r) Method ••••

Use of Check Terms. • • • • • • • • • •• • • Abridged Method B for the 'Solution of Normal Equations • • •

Normal Equations with Constant and Check Terms ••••• ,Abridged Solution B with Notes on the Process .. • • • • • Use of ,Check Terms shown in Red.. • • • • • • • • • • •

--Back Solution with Notes on the proc~ss •••••••••

-iv-

54 55 55 55 56 56 56 57 57 57 57 57

APPLICATIONS OF T~E METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.

ADJUSTWENT OF ELEVATIONS IN DAWSON AND ST. JAMES TOWNSHIPS,


INTRODI)CTION. An introduction to a thesis on least squares could be drawn

out to considerable len~th in explanation: as the purpose of this tbesis has been an investigation in applications of the method of least squares to tbe adjustment ot level circuits,it will be found in the tody of Ute rCllort considerable space has been devoted to the explanation of tbe different applications of the method,witb a vie. that these investigations may bo of use to eD~iDeers in the profession; so the intr'oductory remarks will be brief.

It is well known amOn~ engineers and otberstbat measurements made ~ith apparatus of " most any kind,chains,rods,scales,verniers, .icrometer8t~tc., and no matter ho~ careful one may be in making measurewelits,there will ~e discrepancies aWOD~ tbe o~servatioDs. If a series of benchmarks ~e COlJuected by lines of levels,~ome of wLicb are check lines forming witb tbe others. complete circuits, it is necessary to adjust tho difference in elevation so "that aU the circuits will close exactly,in order that the difference of elevation between any t"O ~encb.arks will be constant when co.puted througb two or more series of lines by the several routes • . This is illustrated b) several applic~tions of the aethod of least squares to tbe adjustment of elevations in Oa"son and st. James Townships,Pbelps County. Missouri.

It is ullderstood that observations are always Jiade as carefully as possible,tor tbe most pro~able results and miniau5 errors. Positive and negative errors of the same magnitude occur witb equal frequency; tbey are equall, prot able. Errors of increasing lIagnitude occur with decreasing · frequency. Saal1 errors occur flore frequentlYj8re lIore proba~le tbat. large ones. Very large errors seldoll occur;tbey are likely to belong in the class of mistakes rather in tbat of accidental errors. Acci~ebtal errors are systematically lIodified by tbe circumstances of the observations. The c·hief cire.umstances affeeting a set of observations are atllospber ic conditions, tbe skill of the ofiserver, andtbeprecision of tbeinstrullents.

CON TEN T S •

Adjustment by the Method of Correlates. . , Theory • • • • • • • • • • • • • • • • • • • • • • • 58

Sum of the Weighted Squares a Minimum. • • • • 58 Conditions multiplied by2C ••••• • • • • • 58 Summation of Weighted Squares and the 'Conditions58 The First Derivutive placed equal to zero ••• 58 Nomenclature of Symbols used • • • • • • • • • 59 Theoretic Formation of the Normal Equations • • 59 ,DisCllboion: On Methods for Formation of

Normal Equations .• .• ,,!)9

Application of the Theory • • . . . . . . . . • 60 Derivation of Terms for Level Net(Dawson-St.

James Townships l,Te~) •• 60

PAR TV. PROPORTIONAL ADJUSTMENT OF INTERMEnIATE BENCHMARKS ,

FROM CORRECTED EI.JEVATIONS OF AnJACENT JUNCTION POINTS. APPLIED TO THE ADJUSTMENT

OF INTERMEDIATE BENCHMARKS

IN THE DAWSON~ST.JAMES TOWNSHIPS LEVEL NET,


BASEO ON THE LINE OF LEVELS RUN BY THE U. S. QEOLOQlCAL SURVEY 'N 1908 ALONG THE FRISCO R. R •

• ITH PERM~.ENT BENCHMARKS -A- AT ST. JAMES, . AND -E- AT KNOBYlew.

Diagrams showing Adjustment Corrections, Description of Benchmarks,Observed Elevations,Adjustment Corrections, and Adjusted Elevations of all Benchmarks .in the Level Net.

Adjustment of Benchmarks between "An and "I" • • • • • 63 - 66 " " " " n n " 'I and P .... • • .. - - 68 " " " " " " fI P and Q .. '. • · ..

and -

71 " " " " " " " 'I J .. • • • •

" " " " It " " B and B .. • · .. I. 73 - 76 " " " " " " " H and J • ,e I. • • -77 " It n " " " n J and M • • • • • -79 Adjustment of Points N and 0 • • • .. • • • . .. • • • •

-v-

CON TEN T S . Adjustm~nt of Benchmarks b~t,,~en "M" and "N" • • • • • 80

" " " " n " " N and R 82 • • • • • -" " " " " " " P and M • 84 • • • • " " " " " " " E and D 87 • • • • • -" " " " " " " D and G 88 • • • • • " " " " " " " A and B 92 • • • • •

" " " " " " n B and C 93 • • • • • " " " " " " n C and D 95 · • • • • n " " " " " n C and F 98 • • • • •

" " " " n " " H and F 101 • • · • -" " " " " " " F and G .• 103 • • • -" " " " " " " G and 0 104 • • · · " " " " " " " 0 and s. 106 • · •

" " " " " " " N and 0 108 · • • •

CON C L IT S ION · Main Cla8s~s of Probl~ms,M~thods of ·Solution,Appropriate

Applications. 110 Dir~ct Obs~rvations of a sing1t:~ Quantity. • .. ' . .... 110 Indirect Observations • • • • • · • · .. • • • • • • 110 Condition~d Observations · • • • • • • • • • • • • 110

Two M~thods of finding Most probabl~ Values of Unknown Quanti ties • .• 110

Obs~rvation Equations. • • • • • • • • • • • • • • 110 'Condi tion Equations. • • • • • • • • .. • • • • • • 110 Comparison and the Appropriate Application of the

Two Methods. 110 B~st Methods of Solving Normal Equations .• .. • •• • • .. 110

-B I B L lOG RAPHY .•

.Bibliography • .• . . .. . .. .. .. • • .. .. . • .. .. .. • • • • .. 111

I N D E X

Ind~x,Alphab~tical .. .. • .. · • • • • • . .. • · .. · .. .. .. 113

-vi-

APPLICATIONS OF THE METHOD OF LEAST SQ(TARES TO THE Al).JI1STMENT OF LEVEL CIRCUITS.

ADJUSTMENT OF - ELEVATIONS IN DAWSON AND ST. JAMES TOWNSHIPS~

PHELPS COUNTY~ MISSOURI.

ILL U S T RAT ION S • Folder of Plates preceeding page 1

-Plate .A. Map of Dawson and North Half of St. James Townships showing Lines of Levels,etc. • • • •• 1

Plate B.Showing Observed Elevations. • • • • • • •• 1 Plate C. Rearranged. Elevations of Points for the

Discussion in Parts I and. II • • • • • • • • • • 1 Diagram of the Circuit A, B, C, D, E,A ' •••••••••• 2 Diagram of Adjustment Corrections between Points A and D. • 5

" " "" lie " " "A and I. .18 Diagram Showing the :Adjustment of point P ••••••••• 19

" " " " " points Nand 0 ••••• 20 Plat showing Observed EI.,:vations for the Points P, Q,M, N,

R,O, and B ••••••• 21 -Diagram of Adjustment Corrections between Points.! and I.

" " " " " " " " I and P. - -

" " " " " " " " P and Q.

" " " " " " " " .A and D. Plate showing .Arranged Circuit for Adjustment of Entire

.25 .• ~29 .31 ,.34

Net from the Railroad Base by the Method of Observation Equations •••••••••••••••.• .• 40

Plate showing the Selection of Small 'Circuits for the Adjustment of the Entire Net from the Railroad Base by the Method of Condition Equations and

. Correlat i Ves • • • • • • • • • • • • • • • • • Graph for the Adjustment Corrections between.! and I.

- -Diagram Showing the Adjustment of Point P ' • .••••• Graph for the Adjustmtmt 'Corrections between I and p.

- -" " " " " "e " "P and Q •

" " " " " - " " " " "

" " " " ~vii-

" "

I and 'J. Band lI.

• • .49 • • .63 • • .66 • ,. .66 .. • .68 ,. • .71 • .. .73

ILL U S T RAT ION S . f l .' j.

Graph for th~ ,Adjustm~nt 'Corrections b~tw~~n Hand 'J • .. • 76 n n n " " " n " J and M 77 • • •

Graph showing . .Ad,j ustment of PointeN and 0 • • • • • • • • 79 Graph for th~ .Adjustm~nt Corr~ctione b~t"~en M and "N • • • 80

n n n " " n " " N and R 82 • • • -" n " " " " " " P and M 84 · • •

" " " " " n " n E and D 87 • • • -

" " " " " " " n D and G 88 • • •

" " " " tI " " " A and B 92 • • • n " n " " " " " B and C 93 .. • •

" " " " tI tI " " C and D 95 • • •

" " " " " " " n C and F 98 .. • •

" " " " " tI " " 'H and F 101 • .. • -

" n n , n " " n " F and G 103 · • • -" " n " n " " " G and 0 .. • • 104

-1.06 Graph showing th~ .' Adjustm~nt of Point "8. • • • • .. ... • •

Graph for th~ Adjustllt:mt Corr~ctions b~t",=,~n N and o. • • 108 Plat,=, Showing Figur~ .Adjustm~nt of A G~od,=,tic Quadralateral

-of th~: (P).~(q)~ (r), and Givin~in D~tail the Proc~ss

Solution of Normal Equations A.pplicabl~ to 'Level Net Problems • • • . . . • • • • • • • • • • • • '. .Finas.

- Viifi-

The principle trom which the term "leQstsq~a~es" 8ris~s is the following: In .e4s~re.ents of eq~al _re~J$Joft.the aost _robable val~es of observed q~QntJtles are those that render the s~r. of the sq~ares of the resJd~al er~ors a .JnJ.~". The errors that lIIay be corrected,however,by tbis metbod, are only those accidental errors that are governed by no laws other than those of cbance.

The lIIethod of least squares takes first rank as an aritblIIetical process and by its use unitor8ity ' is secured in tbe process of adjustaent and comparison, and the 1II0st pr'obable balanc,iDg of errol's of observations lIIay be expected.

It is believed tbat the .ethod of least squares is not in general use allong engineers and that approximate aethods are lIIore ot'ten substituted for the adJust.ellt of quantities,although far superior results.igbt be expected through the elllployment of toe 1II0re exact aethod. There are those,Do doubt,who although versed in aatheaatical science, lIIigbt find it difficult to utilize this lIethod, for lad. of concel.ltration eBIl patience to carry this process througb t~ completion.

It is recolllmeLded, however, whenever precision is delllanded and subsequent measurements are dependent upon tbe results.

In the adjustment of level circuits, tbe several lines are weighted inversely as tbeir leagths,tbis causes us to state the above principle 8sfol10.8: In obs.rvat Ions ~h.". ~neq~al . 'e.lght s are appls.d,the aost _robable valves of the observ,d q~a"tJtJes are thos, that render the s~* of the a'Jghtedsq~Qr.s of the resJdvol e~ror$ a ~'"'.v.. '

The levels in Dawson and St. James Townships were run un~er .Y superv isior1 by .essrs. E. E. Dec,ker, W. R. Denison:,and F .lJ. Bollow.

In the arrangement of this thesis, and in tb~ coaputations and construction of drawings, I wish to ac~nowled2e'tbevaluable assistalJce rendered by Major CnarlesE. Cooke,Pro'essorof Topographic Engineering.

-x-

I r7.~ I

I I

I I

I

I I

I I

I I

I I

I

I I

).

------1----. r... ...

PLATE B. f! Showill!L"".e/ Li",s and Ih~ I

direcfil1f1.S in which run. £lertlrilJhs I ~ \ . on: in bl«k d jc!.".,inf.s. IJiffer4ln&44 of' _levoH_ OrtJ In r4d anJ brIJwn iMf_n /HN'nts. I

IJiNlI1l7C~.s ore in gl"4~n. ThtI YO/II~.$ ""ffiis ~ I "',.. th.o >,.,1",.. D<4 IIJI,I,..u",.,,-I.11y ci.t.,..",i"M:I o.u/ co/'"

"'~I'0"" 10 ~ 0";9" ... 1 J.41y.1 N.III.s • .u/ D.6C";I''',.n Books . L "'ilJs W4lr~ /""" alNlg l7isco R.R. holh -fs fl'bm I

I J..,lHItlitJ"'o ,q,.li~'-. 0 tIJI.1 dillwlfC • • .,39 /rO;I ... i" '90~ . 71ri. Ii .. i~ I _~~.::.,!,~~~/~~~~~~~~~!~h~a:~:._"!..~~'!.~~. ___ _

I

This P,-,. /1 • /Y'."""".«- I ,...",1 .f III. .,..,,-".1 d.l. I on ,.".,k B ,,~i"t fit. "il'- I fer."" ••• 1."' .... h ... ~1 on .. " .. ",11,-1,.;,., 4._ I & ........ 1",.'., .,..,.. t;;n.uiUl. J..K.I.P.H,J. I H.F.6,X.nt.A. Wif/r I til. excllpf,_ .; II"" ~ ~. C.D.~J tIM IUffirll.oc.t._. I. -'.!· ... kJ _ /I ..... 1.. •

~ I ~L"I

~--J;=P~~' , I ~~e=U---- I

I I

I , I • I I

I • I

I

I •

,,& ~- 1" " C. I ;s1',II~} "",t:lrr~m!l.~nf of Jo • .".1s i n

131..,18'0 it' . ' circuits for ri9tJroU$ (4)1161- I . menf. Line A.B. C.D Dna' £ toD ;& I

..",O!!!~~~¥~::::-::9J>' oo/usrd hdWNn~ ond £ on t""Dilrood lin.

;'1r,..,..,.$ itNIN:(lt. d'-,..cT- I ion tJfle .. l.silldrcuib. I

J"Mh __ I""inf ele"..!,#n.s I .. re ;, ilMdr, e/lfflll'enG&3 of ele,./,im /lnJ in retl I .nd bl_", c;los .. ",,,,,,, I /" bl_. tt¥1"k>_~ Oi"e. I /n 9re.". D/6IwJOU I:' n.ll co".,."'" "';,." .,J,,, .. ,, i" bu.Lin • .s F'Q.,Nfl. 0

OSOQ~e.",.~

hI ~" o<{;· .. n-...I

• -Ci~~",~ IJT /t!'~/s,fir.sl;ond flJd wll./e ~ h ~ I I 1-a.IS$ north is 1J1~e/ onll,B.C,DQ.$ a,BP-$ • .

J _____ ..: ____ ___ _ ___ _____ __________________ _____ __ •

&I. l. e. oft h •

No,.th.a.t po,.tlon of PHELPS Showln, atlon of L EV ELL I iii E S, BENCH IIARkS, JUHCflOH PIS, wit h ,. •• p.ct p to aactlon IlfI •• and phy.l.al

'1 •• at u ft •• , • t c 0r-'-'n DAWSON T.p. _ ~ i and vlolfllty. 'V" , T hi. II A P a I • UJ-r .how. tho ' D IRE C T ION 'n :ot"""--;i'f-T-1'-,.--"";"'::::'=~"""~,....,.---F-,.--+-..,---t-----!"-t---.--t:='T---'------7---,

which tho LEVELS •• ,.. RUN. • • • • •• • • 00 In follo.l n 3 tho dlacu •• lo off thl. THES adju.t tho COVER IIAPS ov.,. thl. BASE MAP a. t h •

a ,..

• • •

PAR T I • . - -

THE INDEPENDENT ADJUSTMENT OF A MULTIPLE LINE. METHOD OF LEAST SQUARES APPLIED T0 TEE ADJeSTMENT

SF THE LINE A VIA B~ e~ AND D TO E.

(BASEO ON THE LINE OF LEVELS RUN BY THE U. S. QEOl'QI~AL SURVEY IN 1908 ALONS THE FRISCO R. R.

WITH PERMANENT BENCHMARKS~·A· AT ST. JAMES, AND BE" AT KNOBVIEW).

THE O~TERW ' INATIO~ A~O AOJUSTMENT OF ELEVAT10NS IN

PHELPS COUNTY MISSOURI.

(rawson Township L~v~l N~t)

Adjustm~nt of lin~s 'A, B, C, I; and E,r as Multipl~ Lin~s of 1~v~I~. S~e sketch below and Plate C.

-~ULTIPL F LI~KS. By a multiple line of levels is'

meant a set of two or more lines connecting th~ same two b~nch marks by routes of differ~nt length. In ord~r to find th~ most probabl~ value for th~ diff~rence of el~vation between th~ terminals of a multinle line, it is n~c~ssary to wei~ht each constitu~nt line inversely as i ts len~th. Havina weiahted the sev~ral 1 ines , as thus explaned, the caSe becom~s identical with any case of weiahted measurements, and hence the probable error of a sinQle measurement of unit weiaht is aiven·by the formula:

the pr()bable error of any of the linea of the wei~ht "J! by the tormuls:

and the probable error of the wei~hted ~rithmetic mean by the formula:

/ ZE V 2 = 0.6745 \(Zp(n-1)

Our problem is as follows: G I .;. n two I In •• o f 1 • .,.I,·,a, ,hown I n til.

a" 0 v. f I 3 u .~ • , • h I 0 h l I v. t ". f 0 I low I n l ~ •• u It,:

A v I a B, 0, to 0, 6. 25 .1 I • I In. , + 1 1. 1 1 1 ft.

A via E to 0, 1.00.11. lin., +11.049 ft. Not.: A to E llv.n •• 13I1t of Inflnltr.

T h. • I • vat 10 n of 8. It • at A 1017.800 ft .. , and the .I.v-

atlon of B .... at E • · i074.001j ft.

Rft'aui ~ e 1 t: he mos t orO b lblf! s l e v at io'l of

the 00 " t ') , ano tha proba~l .. e rr o r o f tni 9 result.

SOL UTI 0 N

· M P pM V V2 pV2 11.111 o. 160 . 1. 77776 · +0.0 53 0.00'2803 0.00044944 11.-043 1.000 11. 04900 -0.003 0.000081 0.00008100

Ep:l:-160 _ ... _-----) 1! ~. 82676 --------11. 058

O.t.".· ln·atlon of t II I allov. quantltl •• :

A 1087.800 0 . 1098.911

--------+ 11.111

01 ff.EII • . ,. C t 0 D.

ft. ft .•

ft. • ,. a 8,

A • 1081.800 ft. E. 1074.005 ft.

1-3 .. 795 ft Dlff.EI.v A to E.

D f~o. O~ark H-1 lin. • 1098.911 ft. o f ~ o. K" 0 " • I • w 8. • III. ' . 1 0 9 8· . 8 " 9 ft.

Ob •• ~v.d .rror at D .0.062 ft.

----------Ep~O. 00053044

n=!2.

D 1098.849 ft. E. 1014.005 ft.

• 24.844 13.195

• 11.0 4 9 ft. Dlff.EI.".A via E

to D.

RESIDUALS: In oba.~.,atl · ona of unequal •• Illlt,the dlff.~.nc. b • t • I • nth e w. I lilt. d a" I ::t:h •• t I 0 •• a nan d t". 0 b •• ,. II. d "a I u • of eaoh obs.~vatlon I. a ,. •• idual. If the ob,.,..,.d "alu. I. 3~latl" tllan the .el ~ht.d a~1 th •• tic .ean, ~h. ,..aldual quanti t yl. p I u •• , and If I... t han , h. •• I :I h t • d ." I t h • • t I 0 ••• n , the ,. •• Idual quantity I, .Inua.

FI~.t o",."".d dlff.ln EI.v II ,.Il"t.d .~Ith •• tlo •• an pli

11.111 ft. 11.058

FI".t " •• i dual .""or •• 0.053

s • con doll •• r w • d d If , 1. I n E I ." II 1 1. 0 4 9 ft. Wel]hted a,.lthlletlo lI.an pI! .11.058

S.co~d ,.e.1 dual ."'0" -- O.OOg·V.

GE,\rEriAL PliIN'CIPL-E: In ob·serV.:Jtl.071S of unequ.:JI />rec~s.orn,

the mast probab!e values of the observed quantities are

those that render the sum of the ~elRhted squares of the

res. ~ d u a I err 0 r sam I r, I mum. I t I. 0 n a coo u n t 0 f t hi. P r I n _ clpl. th.t the lIethod of least .quar •• has 1I •• n so na.ed.

THE WEIGHTS P are the reciprocal's of the distances for each line. Thus:

A via B, C, to 0 6.25 rall •• ,r.clp,.ocal or wt.-0.160.

E to o 1.00.11. .. .. .. .. If • 1. ·0 0 •

N.o t e: R • c I pro 0 a I san d S qua ,. • .• can b. If.' Y oon".nl.ntly look.d up n Bar'ow'. Tables of Squar •• , Roots, E~o.

Nth. nUllb.,. oft oll.ervatlons.

CHECK: A. a oh.ok on the solution .0 ... i ] h t • dr •• I d u a Is", U st. q u • I z. ""0 •

p o. 160 1. 000

V +0.053 -0.009

pv +0.009

:~!.~~2 l:p v - o. 0 00

far, the s U II .. a t Ion

Then the probable error of the weiQhted arithmetic mean is:

r = 0.6745 IJQ. OQ~53044 =': 0.0144 ft. p a V 1. 1. 0 x 1

1087.800 · ~ 11.058 = 1098.858 ft., And the most probable value for the elevation r is:

~ = 1098.858 ~0.0144 ft.

INTERMEDIATE POINTS. By an Int.r •• dlah point i. "alit on. Iyln] only on a sln31. In. of I.".,., and h.n •• h a If I· n l n 0 I n flu • no. 0 nth. l. n era I a d j u • til. n t. T h ••• • a y ooour on a line whoes .nd. half. b •• n tatl 'faotorl Iy ad-

o f

justed or on a olos.d olrcult. In .Ith.r oa •• ,th. r.qulr-.d adju.tlle"t Is dlstrlbut.d uniforlll), throulhout the 11".,lIallln3 the oorr.otlon b.t· ••• n an)' two polntsdlreotly proportional to the I.n]th b.t ••• n those t.W points.

ADJUSTMENT OF INTERMEDIATE JUNCTION POINTS B & C.

Ollser·v.d valu. of D - 10 9 8 • -9 11 ft.. (Ozark H.'1. LI".) Adjusted val ue o f D - 1098.858 ft.

--------AdjYstllent • + O. 0 5 3 ft. 10 •• r at D and tap e r to 0 at A.

-4-

A. lhe dlatance A "Ia tha dl.tanc. A to B 1.00.1

B, . ; C to t h • n

o • t h •

6.25 .. 11 •• , and adjuat •• nt to

poInt 8 I. aa foll .o.a:

Adju.t •• nt t 0 8 0.008 ft.

And the IIO.t probabl. el."atlon of point 8 ia:

1074.666 0.008. 1074.658 ft.

In like .. ann.r, a. the dlatanc. A to C" 4.00.11 •• ,

tit. a d j u.t •• nt to C - --~-- " O. 0 53 - o. 034 ft. 6. 25

II n d t h • .0. t p rob a II I • • I • " a t I on o f pol n t C I • :

- 1083.400 ' - G. 034 • 10S3.366 ft. - -- - - - - -

• • • • • • • • 0 • • •••• o ••••

C H E C K 0 n • I • " a ~t : Ion 0 f 0 fro III E •

EI."atlon of 0 from E • EI,vationof E

Olff. In EI,,,.E to 0 Corr,otlon for 1.00 .. lIe E to 0

109S.849 ft. 10~"oI00!) '

.24.S44 ft. -.0.009

24.853 ft. 1074.005 --------

Mo.t probabla E I a " • of 0;'1098.858 f t

Ob •• rv.d EI. v. 0 t 0-1098.849 f t Adju.t.d E I • t. of 0-1098.858

--------Adju.t •• nt. o. 009 ft.

o • • • • • • • • • o •••• ••••• ( 0 • K. for

• •• e .e

S •• Plat. C for the .. dju.t.d " .. Iu •• for this clroult •

• g~

~.O~~~~=+~ __ ~ ________________ ~-L ____________ ~D~-L Adju.t.d Llna

10-------- 4: OM;. --- -----01 ~-------------6ZSHi----------------~

DIAGRAM SHOWING ADJUSTMENT OF POINTS SET. A .. nd O.

t I ..... 1· ' a •• n how the ,v .. lu ••

1 "".).

Fro. t •• abova dl .. ,r ... I alon3 the OIl •• r"adtr.d)1 to the dl.tanc •• fro .. II T~~·. 16" .. 3 r .. .. a I • 0 • a r v ••

In • .. r. lo •• rad proportloll .. 'I, do. n to t'll. .. d j u • t • d , ( II I .. c k .) I In,.

tit. In t • r •• d I • t. p ·o I n t • •• .. oh.ok on til ... dJu.t •• nt. ~f

If dr •• " to .cal • •

-§-

PAR T I I .

TllE LEAST SQUARE ADJUSTMENT OF A LEVEL NET

BY METHOD OF OBSERVATION EOUATIONS. APPLIED TO THE ADJUSTMENT

OF THE: DAWSON-ST. JAMES TOWNSHIP LEVEL NET~


(BASED ON THE: A VIA B, C, AND D TO E BASE).

-6-

THE DETERMINATION AND ADJUSTMENT OF ELEVATIONS I N


(Lawson Township Level Net)

Referring to the :Cawson Townshi p Levt::l Net I:iagrams indicated on pages and , the field notes show the follow-ing resul ts:

A via L, K to I = 25.875 ft. -

I via P to M = - 107.201 ft. J to H = + 73.460 ft. M to J = + 13.695 ft. H to F = - 40.447 ft. F to G = - 80.066 ft. G via X, L to .A = -+ 166.589 ft. I to J = - 93.531 ft. A via B to H = - 45.750 ft. A via C to !i' = - 86.657 ft.

-M via N, 0 to ,.., = - 33.201 ft. I-:!

It is desired to find the most probable values for the elevations of the level net junction stations I, M, J, E, F, and ~ Opon finding these junction elevations, it is then desired to find all interme~iate and remaining elevations in this net. Referring to pageZ,it will be noted that the line of levels running from A via B, C, :C, to E has previously been adjusted; levels along the Frisco R. R. have also been adjusted:these lin~s will be considered base lines and their values will remain unchanged. The :Cawson Township net will be considered tp hinge on the O. S. Geological SurveyStaf;ions A (at St. James) and E (at Knobvi~Hd.

SOL UTI 0 N 0 F PRO B L E M BY MET HO~ 0 F lEA SoT 8 0 U ARE s.

As there are but,six unknown Bench Marks (B.Ms.) · (I, M, j,sH, F, and ~),there can be but six independe~1 unknowns ih the observation equations. As the linesAI~IM,

- -MJ, JR, HF, and FG, may .evidently be selected as the indep-enden1t' unknowns, we may write for . the most probable values

-tr--

of the correspondinQ differences of elevation:

A via L, K, to I = '25.875 + VI -I via P to M = - 107. '201 '+V2

.' M to J = + 13.6:15 + V3 J to H = + 73.460 :+ VI,. H to F = - 40.447 + V5 -F to G = - 80.066 + V6

The conditional equations involved in the se'veral closed circuits may then be avoided by wri tin ~ all the observation equations in te'Ilms of these Quanti ties. W ritin~ the reduced obsetvation eQuations directly from the dia~~am on pa'(1e1C, we have, by ,comparison with the observed values,

R ° \I t. ' VI R • • I d u a f • g'5:~6~·(wei~~l3~:\60); A Vla L,K,toI =

-I via P to M = V2 = 0.000 (wei £!h t 0.156);

M to J = 'V3 = 0.000 (wei£!ht o. 31'2) ; - 0.6:25); J to H = VI,. = 0.000 (wei!!ht H to F = 'v 5 = 0.000 (wei f,1ht 0.571);

-F to G = V6= 0.000 (wei ~ht '2.000) ; -

(wei~ht 0.222); G vi a X, I, to A = - 'V 1- V 2- V 3- V I,.-V 5-V 6=:+0. 155

A via E to H = 'V 1 +V 2 +'V 3 + VI,. , =-0.171 (wei Ith t 0. '2:14);

A via C to F = 'v 1: +V 2 + V 3: +'V 4: +V 5 = +0.'28:1 (weiltht o. 333); - V3:+'V4+'Vfj:+lV6=-0.157 (weiltht 0.1:24); M via N,O, to G=

I to J = V2+V3 =:+0. 0~ 25 (wei~ht 0.6'25). . ,q .. atlo".

To show how the above reduced observatlonAwere form-ed, a compilation of the valu'es for elevations of the several level net junction points determined from the suc6essive differences of elevation alon~ the selected circuit, A via L, K, I, P, M, j,B, F, G, X, 'D,to A 1.s shown below.

1

The closures shown above for the several circuits are extended below, as well as the calculation of the respective

wei ~hts.

-2-

Observed values for the elevations of stations alon2 the Circuit A via L, K, I, P, M, J, H, lI, ': , X, I, to A, Ietermined from successive differences of elevation:

A = 1087.800 ft. U.S. S.S.Iron Post at St. James. 2.5 .•. 875

--------(-I = 1061.925 ft. --------1 = 10.j1.9' ~5 ft.

M =

J =

it =

=

=

A =

+

+

-;..;.------- , I I I ,

--93~506 ft. 9 3 .5 31 ft. I to J. '

I

,-------- , ' ~ 0.0:25 ft. Closure. 968. 41'j ft. -------

I

73.460 ____ '_-"__ A =1087.800 ft.

1041.87) ft.---- ," ' ~ . ~:T: ----------40.447 45.121 ft.

-------- 45.750 ft.A 1001.4~2 ft. --------

, - 0.171 Closure. "" 80.066

viC!. B to B.

912l. 366 ft. , " A= 1087.800 ft ,

', ____ 1

+ 166. 589 " " ---86~368- ft. .. ' ".

" 1087.955 ft. ... 86.657 ft. A via C to F.

'" ---.-----"'~ 0.:289 ft. Closure.

" 0.155 ft Closure. "" M = 954.7:24 ft.

" , " ,, - ... ~ " ,--- . : , ' -33. 358 ft.

-3;.201 ft. M via 0 to G.

A . ROt r *. Vla . , , to ..

I via P to M

M to J -J to H

H to F F to G -G via XiI, to .A

.A via E to H -

M v iaN, 0, to G

I to J ' .A 'vi a 'C to F

= 3.21 = 1. 60

= 1.75 := 0.50 = 4.50 = 3.40 = 8.05 = 1.60

= 3.00

--6:I~~-ft. Closure •

" 0.156 1-- .. ----

= o. 31:2 It will be noted

= 0.6:25 that in each case

= 0.571 the respecti ve

= :2.000 wei ~h ts ar~ the

" " "

O. '2212 reciprocals of the = " = 0. ':294 distances.

" " = O. 1:24

" = 0.625 . . If = 0.333 -9-

We next write the normal equations,which are ari ved at as summarized in the followina rule:

'R {j L' E: 10 for 111 the . nor 111 a l . e qua t ;. I) 11 j 0 rea c;, 0 n e 0 j

t h; e u n ~ n 0 it n qua n t ;. t ;. e' s, 111 U l t ~ ply e a C hob s e r vat ;. 0 r, . e q 'U ,:1-

t;. onby t~e product oj the ~ elght oj that observatlon

and the algebralc coejf;.c;.ent oj that unknoanquant;.ty

.n that equat;.on, and add t h e results.

Bavin~ formed t~e several normal equations,t~eir solution as simultaneous equations aives the most prob~ able values of the unknown Quantities.

Then by the above rule; we have: NOR II ALE 0 U A T ION I NV 1 • (Solution f 0 ,. )

O. 160'v 1

O. ~9 4V 1'+0 .2) 4'V 2'+0. ' ~9 4V 3 +0. :2) 4V 4

O 2A~V +0 3 A"A'V '+0 ?: A ~V '+0 ~ ;z :zV '+0 ~?:7V • :"'.-1.-1 1 • -'.-I 2 ' • .-1 -' .-I 3 . ..).-'.-' 4' • .-1 .-1.1 5

NOR W ALE Q U " T ION I N lV % •

'+0. 156V 2

=+0.000000

=-0. 050!L7 4

=+0.096333

=' +0. 000000

0. '22'2'V 1:+0. ' 2' ~:2V 2 +0 . :2'22V 3+0. '22: 2V 4:+0. ''L''L- 2V 5,+0. '2:22V 6=-0.034410

o .~) 4'V 1 +0. '2) 4'V 2'+0. ' ~94V 3+0 .'2) 4'V 4 =-0.050274

0.33~V1+0.3 3 3Vi+O.333V~+0. 33 3V4+0. ~ 33V5 =+0.096333

:+0. 6:2FjV 2:+0. 6'25V 3 , =:+0. 0156~5 II') ~~-;~~j' ~-:-l - ~~~~-:~-~~l~-:~-;~I ~~-:~-~~~~-:~-~-~~:::~-~~ry=ry~ , L, . ' " , .J 1 .", " . ' , " , 4 . , . ' 0, ' , , ... .,

NORWALEOUATIONI.., V3 •

'+0 ?:]/r 'v ' • .1 L 3

O , ,- , ,~ ! r v '+0 II',r ,r>V '+0 ,rr,r,v :+0 ,rf ,rv ·+0 ,rL;r,'.2V +0 ,,~r r;.v =-0 034410 • , L, L ' L l ' • , [L.· L 2' • ,L. L , L. 3. , L L.L 4 • ' L 5' • L L 6 •

, 0 . 1 ~9 4V 1:+0. '29 4'V 2:+0. 129 4'V 3:+0. '29 4'i 4 =-0. 050~7 4

O 3AA'V ' 0 ?: A ~V '+0 ~~:z'v '+0 ~~~'V '+0 3;Z~V =; ~. 0.0963~~ • .-I.J l ' + • J .J .) 2' • -' -'.-I 3' ..) -' -' 4' . , .-'.-1 5 - -

, +0.1' ~4'V3+0. l' 2 4V4+0.1'~4;V5: +0.1'~4'V6=-0.019468

:+0. 61~5V 2:+0. 6·r;.fJV 3 ='+0. 0 156-r;.5 ~) +~~~4?V~:1~474V;:1 ~)lnv~:~~77?v::n~~71V~:G~?~7,v6::~~007807,

-1@-

NOR MAL E QUA T , 0 N 'N 'V 4.

+0.6~5V. =+0.000000 O ";, ':;'S-V +0 ,I""L ,I""L' I')V '+0 ,,:,I""'I""LV '+'" ,I"" ,r ,rv '+0 ,r ,r ,rV '+" ,r .rr,v - 0 0;:14"10 • "-"-"- 1 • "- ' 2' • 4 ' L ~ ' -J. L 'L.L.' • L,LL 5' VO LL.£ 6-- • J "'i

=-0.0.50'274

NORMAL EQUATION I NV 5 •

+0.571V 5 =+0.000000 O ,f",r:l'"'V '+0 ,I"'\""'I"'V '+0 , ""~ '"'V +O· ir"irroV +0 f"""""'V ' 0 ,(''>t')2V 0 0::'<:4410 • ,L·£,L l' • , LL>L 2' 0 ·L.L.·L, J o , ,L,LL... ..L.LL 5 + • ,.L..L 6=-' ~

O. 333Vi+0. 333V;+0.333Vi+0.3~3Vi+0.333V5 ~+0.096333 :+0. 1' ~4'V 3:+0. 1':24V • +0. 1"24'V 5+0. 1'24'V 6=-0.019468 ---------------------------------------------------------5) - ~~'r " ~ 'r'-'] ~ "1"" "7 " /h,"', . ,. -,,- , '" " '/ ' ) o"\~~,-'r "'" , .'. : .... r ' ' f , "'" r ~ - , ~ ':' h " .. ' ' ........ .' i ' , " .J. ,' : ......, ( -i ; .+ ' ' ''''': 1, . +. ' .. ,--' I, ~ -:::: .. ,-... , 1 L.I.. _.' , ' ~": f'

• • ' • ' I 1, . ' " ' :> ,' . . "") • .' ~.1. - .. 'i'" , ",- .. ' :. , ' .

NOR MAL E QUA T I 0 Ii 'N 'V 6 •

: ~2.000V6~+0~000000

0. '22.LV i+O. !222V 2+0. '222V 3:+0. '222V .:+0. '2'2:2V 5+0. :222V 6=-0.034410

___ ~ ___________ ·~~:~~~'~i~~:~:~~·~!~~:~'~~~i:~~:~!~~'~!::~:~~~~~~ 6) .. oJ, ") ./~ ,~ \I + '" -: r: ':' ,: .. ') 7 ~ r \ I . + .'" ;: .A ~ ,;,' + -, 7' :: \i + r;, ,' 1 t.:) 'I • -= _ .~ n ;~ 7. q 'y:~

. • .t .~, , ' 1 , .,' ..' , ~ ? - ' . ~ ") ,I . 1 .. , ,, . .. . ' I ) "Ii " ' . ." V ") I. . . , .

Wh.nc. fro. tit. aOllltJlon for the Nor.al Equationa, w. writ. tit. el" foltowln3 nor.al .quatlo". tllat ar. to . b. • 0 I, " • d • I .. II , t a" • 0 u • ',/ f 0 ,.. til. "a I II •• 0 f 'V 1 ,V 2' V 3 ,V. ,V 5' & V 6.

NORMAL EOUATIONS.

1):+1. 009V 1:+0. 849V 2'+0'. 849V 3:+0. 849V .:+0. 555V 5:,"0. 222V 6=:+0.011649

'2 ):+0. 849'V 1:+ 1. 6 30'V 2:"'1. 47 4V 3:+0. 8'49'V .+0.55 5'V 5"'0 . '222V 6 =· ... 0. 0!27~27 4

3):+0'. 849'V 1;+1. 474'V 2+ 1. 910'V 3:+0 • 97~'v+0. 679'V 5 ,+0. 346'v 6=+0.007806 • 4)+0. 849'V 1:+0. 849'V 2.+0.97 3V 3:+ 1. 598'V .:+0. 679V 5:"'0. 346v 6=-0.007819

5) '+0. 555V 1:+0. 555'V 2:+0. 679'V 3:+0. 679'V.+ 1. !250V 5 +0'. 346'v 6=;+0. 04!2455 6) ... O. 1222Vl:+0. '222V 2· ... 0. 346'v 3:+0. 346'v . :+0. 346'V 5:+!2. :46'V 6=-0.053878

• • • • ••• • •• • •• •• • • •• •• • ••••• Now w. ar. to ao'". til. abo" •• quatlo" •• I.ultan.ou.lr~

SOLUTION OF NORMAL EQUATIONS.

-11-

SOL UTI 0 N 0 F ~ 0 R 101, 1. L E QUA T ION S.

FI ,.at Sat.

I I I I .. I n a 11 e tV:5'

1 ) :+ 1. 0090V 1+0.8 490'V 2 +0. 8490'V 3+0. 849o"V • +0. 55 50Y 5+0~ '2'220V 6= +0.011649 i ll. 5.1 6 12 .

~ 2) U.l1~90V1: +0.9390V2~ +0.8490V:5+0. 4890V.+0. 3197V5+0.1'279V6=+0.015713 7 ). +o:-~205v~:O:09 OO'v;---------:~O:"§ 606,v::~c5:;2 3 53V~~0:0941V~;:0: 0040 64

!L) +0.8490 V 1 +1.6 30V 2 :+ 1. 47 40V 3'+0.8 490V • +0. 5550V 5 +0. ;2220V 6=:+0. 0~ 27'27 4 -0·17..113

3) /+. t)IJ9'2V 1 + 1. 137 5V 2 + 1. 47 40'V 3+0.7 509V .+0. 5~ 240V 5+0. '2670V 6=+0.0060'24 8 ) : +O:19~8'V~·~O:4}25V~---------:~O:0981V:~0~0310V~:6:0450V~:::O:0'2i250

3)+?i 8i~~XJ+1. 4740VL+1. 910'V 3 +0. 9730V. +0. 6790V 5+0 • 3460V 6 =+0.007806 4)/1.;bbbVi+1.6666v2+1.910V3+3.1369V.+1. 3~29V5+0.6792V6 =-0.015349 9 )-o:8176V~:O:19~6V~--------:'2:1~3§V::O~64~9V~:6:333~V~-:::O:O'23i§5

4) + 0 • 8 .9 0 V A + O. 8 • 9 0 V 2 + 0 • 9 7 :5 0 V 3 + 1. 59 8 V • + 0 • 6 79 0 V 5 + O. 3 .6 0 V 6 . • ,- o. 0 0 7 8 19

5)/d!7~~~~1+0.79 5 3V2+0.9730V3+0.9730Vt1.791LV5+0.4958V6 =+0.0608~6 10) +O:O.537v~:O:0537v~---------:O~6250Vti~lT~'2v~:O·:1498V~-::O~068655

5) ·+0. 5550V 1:+0. 5550V 2+0. 6790'V 3+0. 6790V o4+1. :250V 5 +0. 3460V 6 =+0.04'2455

6)/d~~~~i~~+0.4357vi+0.6790V3+0.6790V.+0.679V5+4.6039V6 =-0.1057~L 11) : +O:1193V~:o:1193V~;:--------:--------:O:5710V~:4:2579V~:O:148187-

Su ..... ,.y of fl,.at .et of equatlona,

7)+0. 5200V 1 -0.OjOOVi+0. 3600Vi+0.235~V~+O.0941V&=-0.004064 8) +0. 19 38V 1'+0. 49'25V 2:+0. 0981V 4+0.0 310V 5-0. 0 450'V 6=:+0.021250

9)-0.8176'V1-o. 19'26V 2- '2. 1i)39'V.-0. 6439V5-0. 333'2V6=+0.O'23155 10) +0.0 537v 1+0. 0537V 2:+0.6 250V .-1. 11':22V 5-0. 1498'V 6=-0.068655 11) : +0.119~Vl: +0.l19· 3V2 '+0. 5710V5-4. '2579V6='..-O'.148187

• • • • • ..... 0. , ..... • ••••

Second sat.

1111 .. lnate V 4 • "

7) '+0 . , 5~~OOV ~ -0. 0900V 2 +0. 3600V . :+0. :;: ~5:3V 5 +0. 0941V 6=-0.004064

8)/o:7111~~1: +1.8073V2' +0. 3600V.,+0.11 38'V5-0. 16 51V6=:+0.0779812 1!:;:) - 0: 191:2v~:1:89.73V~---------::o:1!2T§v~::o :~59\2V~::O:08:2046

'8) ;+0. 1938V 1:+0. 49!25'V 2:+0. 0981V . ;+0.0' 310V 5-0. 0450V 6 = +0. 021!25O'

'3) /:..~ b ~'i'v~!o. 0087v 2-0. 0981V .-0. ()29!2V 5 .... 0.01~lV6:::+0. 001050 13): +& :I567V~;:O:4838V;---------:~o:o618v~:O~060Iv~~:o~O~;~306 , '

-12-

SUMMary of second set of equatione:

1':;;)-0.ljl~V1-1. 8')7~v2 , +O.1~15V;+J.:;:5'1:2V6==""0. 082046 13)+0. 1567v 1 +0. 48 ~8V 2,+0.00 18V 5-0.0';01 V, == +0.0':;: L 300

14)-0.6~17v1-0.0067v2-4.4946V5-0.851~V6=-O.;14545 11):+0. 119 3v 1 +0. 11'1 3V 2 +0. 5710V;- 4. :2,79V 6= +0. 148187

••••• • • • • •

Tn i rd set.

14) - o. 6311 V ~ - 0.0061 V 2- 4. 4946 V ;- O. 8518 V 6. - O. 214545 11)!~!~~l'~i+0.6~17Vi+3.0:;:35V -2~.5458V=+0.784658

-------------------------j---------~---------17)+0.6250V2~1.4711V5~23. 3'176V~+O.~7011~ ;

su •• ary of Thi rd set of equation.: , '

15)-1. 3070v;+0.t~37v~+0.185'1V6=-0.054836 16):+0. 48!21V 2-1. 1131V5-0. '2714V6=-0.030'1'20 17)+0.6!250v 2-1. 4711V5-23. 3'176V6:+O.570113

• • • • • • • • • • • • • • •

Fourth set.

I I , I • , nat .V 2 •

15)-1.3070VI+0.]:237v~+ 0~1859V6=-O.054836

16)/i~~3~6~2-3.0177v - 0.7358V 6=-0.083B26 18) --------:~:8340V~:-O:5499V~::o:13866~ 16):+0. 48'21V g-l. 1131V 5- O. '2714V 6=-0.0309:20

17) 2~3~:~i~!:~:~~~Z~2:~~:~~~~'~!:;~~:~~~2~1~ 19) :+0. 0:21ov ;:+ 17. 776oV6 =-0. 47068'2,

-13-

summary of Fourth set of equation.:

18)-'2.8940V -5

19) +0. 0216V '+ 5

O. 5499V6=- 0.138~62

17.7766V 6=- 0.470682 •••• • • • • • • • • • 0 ••

Fifth set.

I I I I .. I nat eV 5 •

18)~2.8940V~- O.5499V6=- 0.13866~ )(13~·9.8148

19) /.3:'~~~~~.2~3l~~:.2~23~.2::§~:.~~'3~7~ 20) :+:2381. 185 3v 6 =-63. 201333

summary of FI fth Set of equations:

20) +2381. 1853V 6 =-63. '2013 33 ThenV 6=-O.0'2654

• 0 • • • • • • • •

Substl'tuts this va'ue fo,' V61n equation IS) and sol". for V5:

18 )-'2.8940V 5 +O.014594346 =-0.13866'2 Then -2.8940V5=-O. 153256346 And V5=+0.05'2956 subatl tut. theT,,,,lu •• found for V6 and V5 In equation 15) and ao\ 've for v2:

15)-1. 307ov 2 +O.0065506572 -0.004933786 =-0.054836 Then -1.3070v 2=-0.0564528712 And V2=+0.0431927

Su b s tit ute the val ¥.,. .. f 0 u n d for V 6 ,V 5 ,a n d V 2 In. qua t Ion 11) and .olv. for VI:

11)+0. 1193Vl+0.00515288911 +0.030237876 +0.113004666 =0.148187 Then +0.1193V 1 =-0.00020843111 And Vl=-0.001747

SUD. t I ,u ~. t h. v a J.u •• for V 6 ,V 5 ,V 2 ,a n d V 1 I n a qua t Ion 7) and .0 I v. for V 4: :<:6 0 7)-0.00090844 -0.063887~2~~+8.3t24605468 -0.002497414 =

==0. 004064. Then +0.3600V4=-0.0092313498 And V4=-0.025643

-14-

Sub.titute the In equation 1)

" a I u •• f 0 u n d fV r V 6 ,V '" , 'V 2 ,V , and V and .01". fOT' ..., 1 4 3 • .

1)-0.00176'2723 +0.0366706023 +0.843V 3-O.021770307-

Thtm ·+0. 849V 3=-0.02498667:23 And V 3=-0.02943

-0.00583188 =+0.011649.

00000 00000 00000 00000 00000

'Check these V values thru one of the equations in the first set, and we find: 4)-0.00148~203 +0.036670857 -0.02863539 -0.040377514+

+0.035957124 -0.00318284 =-0.007819 And we find that these values will dhebk 0 = 0 thyu the fourth aecimal place.

Summary of residual values. 'V 1=-0.001747 ft; V2=+0.04319~ ft; 'V3=-0.029430 ft; V4=-0.02564~ ft; V5-+O.052956 : ft; V6=-0.026540 ft.

Now we find that the most probable values for the dif-ferences of elevation are: A via L, K, to T= -~25. 875 0.0017 = - '25.8767 ft;

-I via P to M =-107. '201 + O. 04 ~2 = -107. 1578 ft; .. M to J '= + 13.695 0.0294 = :+ 13.6651) ft; -J to H =+ 73.460 -0.0256 = + 73.4344 ft; H to F -- 40.447+ 0.0530 = - 40. 3940 ft; - 80.066 o. 0~26 5 F to f"< -- - = - 80.09'25 ft. \,;l

- - -Then from A via the route L,K,I,P,M,J,H, F,G,back to A, we have the most stations. A= 1087.8000 ft.

-25.8767 r,i6;i:§!233 ft~

-197.1578

M= -9."54:7655 ft.

probable elevations

M= 954.7655 ft. +13.6656

J= 968:-'431"1: ft. +73.4344

H= i041:-8655 ft

, -19-

of the unknown

H=1041.8655 ft. -40.3940

F= 1001:-"4715 tit. -80.0925 •

G= ~9!21:3790 ft.

(0 ) N 0 R M A L E Q U A T I 0 N S . V1 Va V3 V. Va V, Absolute

Term 1) +1.009 +0.849 +0.849 +0.849 +0.555 +0.222 -0.011649 = 0 2) ~e.~~9~.630 +1.4?4 +0.849 +0.555 +0.222 -0.02?2?4 = 0 3) +~ • .a/4,9 +,1. ftJl"il +1.910 +0.9?3 +0.346 -0.00?806 = 0 4) +9. a~;i) +9. jJ~,9 +XJ. ~7Z + 1.598 +0.346 +0.00?819 = 0 5) +~ • .5S5 +.0 • .5,513 +XJ.Bff~ +0.~7~ +1.250 +0.346 -0.042455 = 0 6) +G.n2' +0.Z7-~ +0'.3:4,.6' +,0'.,3'4".6' +.0. Z~$' +2.346 +0.0538?8 = 0

(Nu.~er. ~ e low t ". red I I ... ar. redundant nu.~.r.).

(p) Solution of Normal Equations. 1 2 3 4 5 r; Absolute

Term P.1 +1.0090 +0.8490 +0.8490 +0.8490 +0. 55pO +0.2220 -0.011649 Pit (-0.991~ -0.8414 -0.9414 -0.8414 -0.5501 -0.2200 +0.011545

P3 • .. .. • +b.915~ +0.?59? +0. 134? +0.0880 +0.0352 -0.01?4?3 P4 ' - · - . . ·(-1.092~ -0. 829? -0. 14?1 -0.0961 -0.0384 +0.019082

Pa ' . · - . . . ' . . +0.5654 +0.1469 +0.1401 +0.1300 +0.016968 p, • ' . .. .- . . ·(-1.?68~ -0.2598 -0. 24?8 -0.2299 -0.030011

p, •. •.• . . • +0.825? +0.162? +0.1202 +0.015?82 • - • • ' . • (-1.2111) -0. 19?0 "';0.1456 -0.019114 P.· . . . . . . . ~

P, . . . . . . ... . . - •• +0.8694 +0.1646 -0.041682 • ·(-1.150~ -0.1893 +0.04?943 PI. 0 • - • . • - • . •. • . .". • . • • . • . • •

Pu Pu

... .. . .. . . . . 0 . 0 .. , . , _,

· . .... ... . . ... . .... . . - .. . - .. ' . . • +i.21?~ +0.058803 · · (-0.450~ -0.026514

(S)Find most probable values for the dif-ferenoes of elevation.

A via L,K,to I = -25.8?5 0.0016 = - 25.8?66 ft; I via P to M =-10?201 + 0.0440 = -10?15?0 ft; lot to J =+ 13.695 0.0304 = + 13.6646 ft; J to H =+ ?3.460 0.025? = + ?3.4343 ft; H to F =- 40.44? + 0.0530 = - 40.3940 ft; F to G =- 80.066 0.0265 = - 80.0925 ft.

(T) Then from A via the route L,KLltP,M,J/H,F,G,b&Ok to A, we have the most probable elevations or · he unKnown atation$:

A = 108? 8000 ft., M = 954. ?664 ft., H = 1041.8653 ft., =_~Q~§Z22 t!~L2~i~ =_~QL~~~Q

I = 1061.9234 ft., J - 968.4310 ft.,F = 1001.4?13 ft., =~ZL!22Q ±2~Lt~t~ =~§QLQ~g§

lot = 954. ?664 ft., H = 1041.9653 ft., G = 921.3?88 ft.

-16-

(q) Solution of Normal Equations.

2 3 4 5 p, Ahsolute Term

q,. +1.6300 +1.4740 +0.8490 +0.5550 +0.2220 -0.027274 q,. qe ,..P,!7.1.43 -0.7143 . -0.7143 -0.4670 -0.1868 +0.009801 ~ ----

+O.~i5~ +0.75a7 "+0.1347 +0.0880 +0.0352 -0.Oi7473 __ ~ ___ ,. ___ o __ -, _____ -.-... .. .- , _" ......- ___ ... ____ , ...".,- ____ . _-_. _ _ _ • __ ~ ____ , _ __ , _ _

q, • • •• +1.9100 +0.9730 +0. 6790 +0.3460 -0. 007'806 q, q.. • . , • . • -0.7143 -0.7143 -0.4670 -0.1868 +0.009801 q.. q, • . • -0. 6303 -0.111~_ -0. 071? -0. 0292 +0~_97~ qs

+~5854 +0.1469 +0.1401 +0.1300 +0.016968 -_. __ ._ ._--,-----" ----:-..... _. -..-. ---_ ...... -.:

.0. • +1.5980 +0.6790 +0.3460 +0.007819

. " • . • •• . " -0.7143 -0.4670 -0.1868 +0.009801 ..... _ . .

Ql0 • .

qll • qu . , Q13 • Q14 •

.. .. ... .... . ... ... .. . _. . ... . .. ... . ..

Q:f 15· • . • . • • . •

Qie • • • . • . Q1. 7· • • • •

-0.0198 -0.0129 -0.0052 +0.002570 -0.0382 -0.0364 -0.0338 -0.004408

+0.8257 +0.1627 +0.1202 +0.015782 --,-----.-~--.. ---. -

. , • • +1.2500 +0. 3460 ~O. 04245.5 • . " . • . -0. 3053 -0. 1221 +0. 006408 • . • • • •• -0.0085 -0.0034 +0.001679 • . • • • • • -0.0347 -0.0322 -0.004205

• • • . " -O~ 0321 -0.0237 -0.003109

+0.8694 +0.1646 -0.041682 -~:--~--------I-. . . • •• +2.3460+0.053878

• •• -0.0488 +0.002563 • •• -0.0014 +0.000671

<118 •••• e_ . o ._ • • . • • • . -0.0299 -0. 00~901 •• , ••••• " -0.,0175 ~O. 002298 Qs.: e • . • • . • . • • • • . •

~o • • . • . • - • . • . • • ·0. -0.0312 +0.007890

+2.2172 +0.058803

(r) Solution of Norm~l Equations. V1 V2 \b V4 V", V6

~ +0.011545 +0.019082 -0.030011 -0.019114 +0.047943 -0.026514

r2 +0.005833 +0.001018 +0.006096 +0.003860 +0.005019 -----~ ~0.029134 -0.005089 -0.013124 -0.010434 +0.052962 -. ---1'-~ +0.021614 +0.003779 +0.006674

ra +0. 025.549+0. 0~5~ -0.030365

. ~ -0.037007 +0.043983 !

ra, -0.001600

-0.025688

AdJustment V" =-0.001' V2 =+0.0440 Va .y:-0.0304

Quantities: V4 =-0.0257 Va =+0.0530 v, =-0.02'5 Cheok ~hese V values thru one

of the equations in (O),and we find: .

L1) -0.001614400 +0.037341567 -0.025779885 -0.021809112 . +0.029393910 -0.005886108 -0.011649000 =

And we find that these values oheck 0-0 thru ~he seventh deoi.al place. . '

-17-

o j

THE DETERMINATION AND ADJUSTMENT OF ELEVATIONS IN


(rawson Township L~v~l Net)

ADJUSTMENT OF INTERMEDI ATE JUNC~ION POINTS L, K, P, H, O.

No1/e:(See discu •• lon on nter.edlate Point. on pa3. . )

to

Ob •• rved valu. of Adju.ted valu. of

'" 1061.925 ft.(R.publlc Rd.Llne) 1061.923

Adjuatll.ntl + 0.002 ft.Lo •• r lin. at and taper to 0.00 at A.

d I at. A. th ... A via L, I(

L • 0.5 ",lleeithen t a I • 6. 25 • I the adju.t",ent

dlatance i. a.

follo.a:

ea,and the to point L

Adjuat •• nt to L _~&.5_ " O. 002 • 0.00016 q. 6. 25

And t h • fa 0 , t pro b a b, I • e I .vatlon ° f pol n t L I , ,.

1078.427 O. 000 1078.427 f-J. --------I n I I k. • ann.r,a, the dletance A to K • 1. 75 • I I e s ,

adjust •• nt to K _1:.12 6. 25 " 0.002 • 0.000150 ft.

And the "'oet probable el.vatlon of point K I a.

0.001 • 1092.693 ft •

.A o,,,;;j ,K Adjuated Lin. I . io----/.7SMi.-----< S 10----------'-----6.2 Mi·~. -----------1

D I A Q R A.. S H 0 'w I N GAD JUS T .. E N T 0 F POI M T S BET. A and I •

Frail ih. above dlagra. It Ie .... 11' •• e" how VII. "at ••• along the Ob •• w-".d IllIe ar. lo •• ,..d proportional to tbe dl.ta"o. f,.o. A to til. Adjuat.d II" •• Tb. dl .. gra. al,o •• ,. v •• a. a 0 h • 0 k ° n til. a d j u • t •• n t 0 f t". I n t .Ir" •• d I .. t e Pointe If drawn to •• al.a horlzontallr and ".ntloallr.

-12-

Obser"ed Adjusted

yalue of yalue of

pol n t pol n t

1061.925 1061.923

ft.(Republlc R 0 ad)

Adjustment + 0.002 ft. II ne to ~e lowered at I.

Arran]ed value of point -_101]-(In lar]e ciroult fro", sU"'Matlon of consecutlye dlffs.ln eleyatlon.)

Adju.ted yalue of po nt 101

Adjust.ent

954.724 ft. (fro,," the

4.0/1,:

0.042 ft.line to •• Il t M.

6.4l1i -

'" e • t )

r a lee d

f~ ~ , ____ -'-c'"'Q"'o~t/on Line --+--- - --.-------1 ~-Io..i

I Adjusted lin e p .-0 ., "'

DI AGRAM SHOWING A D JUS T 101 E N T 0 F

POINT P.

Fro", the dla]ra .. I I n e. I n a n 9 Ie •• juet",ent follow.

;X) ' ,>

<,

Ilcco"'panyln:J ... ty proportionate

the .1",llar trlwe affect the .d

to the point P as

Relations stated ae a propol'tl ,-Q ~' ,

6. 4 0.042·0.002

----------------4. 0 (X.0.002

Sol II e

0.042 + 0.002

x + 0.002 - --------------- x 4.0 Ii. 4

(0.042·0.002 " 4. 0

"

x - -O.0()2 6. 4

And we find:

X -.0.0255 ft. (Adjust .. ent to point P.

Then the .. oet pl'otable elevat on of point P

928.552 .0.026 928.578 ft.

-19-

0.'"

.. ..

(h

m

A,. ,. an J 8 d ( I n I a r' J a SU"'IIIatl on d Iff s • I n

Ad j u S t 8 d

Ar'r'an,ad (In lar'la sUllllllatlon dlf1s. In

Ya I u a 01 pol n t II cUr'cult 1r'olll of consacut I Ya a I • Y a t Ion )

va I u a c 1 pol n t II

Adjust",an

'1alua of polnt-'Gclr'cult fro", o f con. a cut Iva ala'lation).

Adju.tad '1.lua of point G'

t

954.724 ft. (frolll tha .ast)

o. 042

921·523

921.379

ft.Llne to a t M.

ft. (froOl

ba ,..I •• d

III via N;O to

Adju.t",ent + 0.144 ft.Llna to ba Icwa,.ad a t G.

Fro .. the a c COlli pan yin l dlaJ"." by proport on .. t • G; I I n e • I n t h • • I 1ft I I a ,. t r I a n J I a ., • a affact t h a adjust .. an t of pol n. t • N • n d 0 a. folio •• :

r /"

/' A 0 JUS TilE N T OF POI NT N. ... ....

Ra a t I 0 II:' • tat a d a • a P,.opo,.t ..' on «

8. o 5 O. 144 + O. 042 • -------------

O. 9 (0.042 X)

By alJab r a :

O. 9 _x • (0.181) " ----)

8. 0 5 Sol YR l :

x O. 0 2 12 1ft ant to

1t.Adjustpoint N.

Tllan til. 1.1'I.tlcR

'" 0 • t 01

probabla

p c I n t 0' / ' _~" "

j ' ' . '

875.092 ft.

. ·IT' '.: "' I .... , '

I ~ ;-<. I~ ;',, :

01 ' .~.~ ~ M

"

.•.. ~,

". ~ .t:' -. '. ' .

t · ., "

".

., ~o DIAGRAM SHOWING

ADJUSTMENT OF POINTS Nand O.

C on.sfrucfion L /ne,

!.------·a . .3Mi:-. -----..,

/'

.. '

G

1o---------------8·05I'1i.-- - ·- --- --------I ADJUSTMENT OF POINT O. 8.050.144 + 0.042

In tha sa.s .annsr' •• aboya:----·--------~----3. 3 ( .y + o. 0 4 2 )

Solv n3: '.0.0342 ft.Tlla adjust.ant to point o.

Than tn. III0.t pr'obabla .Iavatlon of point

0.034 -969.702 ft.

-20-

G)

THE DETERIIINATION AND ADJUSTMENT OF EL "EVAT'ONS IN


(Cawson Township Level Net) -

Adjustment of lines p~ ani NH as Parallel Luplicate Lines of Le:vels, and the adjust'Tlent of t!lt!

line asa ~s a Suplicate Line of Levels. See sketch below and the three Plates of Caw son Township.

9!J7.Z4-() R t86.,g.~.tJ 9-'/'2..5/61 l.....-- B.M. o4J ", ;S.M. I .-- ~--r- -,- -, --;- -,-1-"-- --,---,- .... -. --r- ..,. -,---,- - 1"'"1

I 1 , I I I I Iii I • I 'I S69:~ d..:::a 758 I

~---~-.--+---I.---~--.:-- +--- --~---;----~-J- -~---~--~--- --: -~- .~ -~'-!--~--J r; I I 1 : I I (t I I : ,96g : I I I

I, I I I I, I;; I I •• ~ • I

[----~----+---1t- · -:~---~--·~- -- ---:---~---I- - ' ,----;---- ---r-- -~---~~ ----:---:---~---I

__ .. : ___ i ____ ~ __ ~ ___ ~. __ ~ ______ : ____ : ___ L _ i~tr. __ : __ L _ _ __ ~ ___ ~ __ ~ __ __ ~ __ -J --J-__ I I : : 1 ~<15 I :: !:: ~ I I: " I : I I' 1

I I I~ I : I I I 11 I' I I 1 I I: I J t---r--r-?- t~.s~ ~ ----t--- --1- --r-- i --- ---;---r--r-- ----1----r---~--I---i---r--r-J ..----:---~---li--,---l' !, - ---- i~--i----:---:--- ---:--T--:--~----T---~--- ;--,

~---:----~--- ---,:---J---L------~---~-J ~J ____ L __ : _______ ~ ____ L __ J __ ---~---~---:'--1 ' I I ,--" . I I I I I' I, I

~: I': : r :MI~ I ,I I:; :': I I . L.: I :. t I , "r t I ' . . .' I r----f----r--'? ---;----r--l---- ---~ ---:---1---- 9'--r- -~---:- -- ---~---:--~--- -----!---i----,---i

: ~ :: I I I:: :: u>: ; • ~ : ::: : :t3 : I

Adjusted values are as ~ 1--p • follows M •

N • I) •

928.578 954.7 66 87~.092 9 6 9.702

ft. .. .. ..

It is required to find the most prcbable elevations of the points Q, R, andS.

ADJUSTMENT OF POI NT Q.

Paralle~ Duplicate Lines o~ Le}els: By a parallel duplicate line of levels is meant a line that is run twice over the same route with equal care and in the same direction.Levels are ~ometimes run in this · manner when there are two instru-

-ments on the work,as was the case when the line PG was run, also the line NR.

Ob.erved .. .. e I e v • t Ion .. .. of poInt P .. .... Q.

Dlf1.renoe In elevation •

Ob •• rved elev«tlon .. .. .. .. of .. point P • .. .. . t •

Dlffere"oe In elevation •

-21-

928.552 1 t • 997.242 .. 1, t 'n.tru.ent. -------+68.69 0 .. .. ..

928.552 1 t. 2nd ItI,tru •• nt. 997.2H .. .. .. .. .------+68.685 .. .. .. ..

{IN ;\'ETGHTEL l!EH SU k ElrfENTS. If U,e d1,jjerence of elev

at1,on oj tmo stat;ons 1, ' S measured a number oj t1,mes 1,n

the same manner,over tAe ' same lenght of l1,ne. arid under

'such cond.t1,ons that tf,e d1,fferent deterrr •• nat.ons may be

reearded as oj equal <l:e1"ght. t fl en the ar.th met.c me<ln of

the 'several re'sults '/, ' s tile most pro b able value of tins

d1,fje r ence of elevat1,on. The probaDle error of a sin)le ... asure"'ent I. ll"en Dy the forllula:

The proDable error of the arlthlletlo .. ean 01 n me •• ure_ ",ent. lIy tha for .. ula:

ADJUSTMENT OF POINT Q.

Ob.erved V.lue.

h8.630 {to 68.685 ft.

2)13Z:'375

v

+0.00:25 -0.0025 -6:6050

v2-

0.00000 ,S25 0.00000625 0.00001250 =L:vz,

z = 68. h875 ft. C hac k n = ~2. (No. 0 fOb s. )

r.= 0.6745 + = 0.0017 ft.

Then the most probable v~l u e for the difference in elevation = 68.6875~ 0.0017 ft.

And the most probable elevation of the point Q is as

follows: Adjuet~~ ¥alue of p~int g ~ 328 .•• ~78 ft.

Adiusted LlfLln elev~P to "" 68 688

Most probable elev. point Q= _~~Z~3§§ ft.

O, b.er"ed • •

-ACJUSTMEN T OF POINT R.

e ley a t Ion • •

01 point N· It f- . R.

875.250 866. H~

Dlfferanoe In elevation .. - 8.867

1 t • • 1. t

• w

O ••• ,."ed ele"atlon o 1 pol n tN. 875.250 ft.

• •

• • • w • • R· 866.293 • 2nd In.tru.ent.

Difference In ele"atlon • - 8.957 • • • •

-22-

ADJUSTMEMT OF POINT R.

The line N to R is a parallel duplicate line of levels.

O~ •• r ... d ... Iu ••

8.867 ft. 8.')57 ft. -----

2z = 17.8'24

z = 8.')12 ft.

r.= 0.6745

v - 0.045 of. 0.045

0.000 C h • c k

0.0020 25 0.002025 --------0.004050 =~V2

n=2. (No. of Obs. )

of. 0.0304 ft.

Then the most probable value for the iifference in elevation = 8.jl"~.: 0·.0304 ft.

And the most probable elevation of the point R is as follows:

875.092 ft. Adjusted value of point N = Adjusted Ciff.in elev.N to R = - 8.')12

Most probable elev.point R = 866:180 ft.

'"

ACJUSTMENT OF POI NT S. CUDlicate Lines of Levels! By a 'duplicate line

of levels is meant a line that is run twice over the same route with equal care, but in onposite iirections.

L.ttln] a r.pr ••• nt the dl.cr.p.ncy ~.t ••• n tn. ,. •• u It obtaln.d from tn. for •• rd I In. ~nd th.t obt.lned from tne r .... r •• lin., •• h .... :

d v = of. ---

1 2 and

d v 2 =

2

Su~.tltutln~ th •• e ... Iu •• in the followln] equatIon .nd r.pl.clnl r 1 with r, for the c ••• of dupllc.t. lin •• , •• ha". the proba~l •• rror of a .'n~l. ",.t.r.'n.tlon (forward or r .... r •• ) by. line of the ""3th 1:

d 2 d 2 '2d 2 (of. ___ ) + ( _ )==+---_

'2 '2 4 d 2

2:~a+ /7.2

Sabsti tuting:

" L ~ o. 6745 y---~~----~: ~Z~2_ y. -n- ___ d ~-1-2(n - 1) 1. 4142136

In the ca •• of duplloa; ••• asur.lII.nts.(n - 1)=(2-:1)= 1. T h • n :

ri.= 0.47f)9 va: =.: 0.471)9 d.

And for the error of the arithmetic mean in duplicate level in~:

r = ",

':0.47i)] d = ----------

VT"

",DJUSTMENT bF POINT S.

Ob,.rv.d .I .... atlon of point 0 • 9119.758 ft(Froa South) "" "" • •• S. 942.516 B

Dlff.r.no. In .I.vatlon -- 27.242(!ioln~ North)

Ob,.rv.d .I .... atlon of point S. 942.516 ft. ." "" • ". o· 969.743 .(FrOIll North)

Dlff.r.no. In el.vatlon - • . 27.227(aoln3 South)

Ob •• r .... d Val u ••

27.242 ft. Then~ r L =':O.47f)j x 0.015 + 0.0071 = '27.227 ft. And:

+ . 0.015 =.: 0.0051 r",=-0.3373 x

------d = 0.015 ft. and 1/2 d = 0.0075 ft.

Then the most probable value for the difference in elevation = 27. '242 - 0.0075 =27. 2495.:!O.0051 ft.

And the most probable elevation of the point S is as follows:

ft. ft.

Adjusted value of point 0 = 969.702 ft. (On Circuit) Adjusted Tiff.in elev.O to S = -27. :250 ft. -------

Most probable elev.point S = '942.452 ft. - - -----SUMMARY OF THE ELEVATIONS OF ALL O~ THE JUNCTION POINTS.

A" 1 () fl 7.f\ :) ~1) ft. '3 ~ :)1 .~ 7 9\) ft. " .. i\ 1 ;; .()9?() ft. B " 1 3 7 .4 • Ii 'i 3· 0 ft. H", 0 4 1 • rl Ii 'i 'i ft. o.. ·~ ~J _ 7 () ? () t t ·. c " 1 !) <i ~ . ~ Ii "(I ft. ,.. 1 n Ii 1 • -1 :> ~ ~ ft. P.. . '~ 2 8. Ii 7 B 0 ft.

" J oq f\ _(l ~ B:) ft. J OI)8.4)li.to Ov" '? 9 1.?lllip ft. ,. 1 () 7 4. I) 01) 0 ft. K· 10 q 2. "i '1 ~ () ft. . R" 8 ~ 15. 1 8 cr-h ft.

F • 1001.471 'i ft. L" 1078.4270 ft. ~ .. 01? 4~20 ft. -t,!. 9. ';i •• 7 1\ ') Ii ft. . .

------~---~------~-----------------------~-.--------------.

THE DETERMINATION AND ADJUSTMENT OF ELEVATIONS IN


(Dawson Townshin Level Net) ADJUSTMENT o F I N T E R M E 0 I ATE TEMPORARY 0.0020 ft.

BEN C H MAR K S BETWEEN THE POINTS ~. I -:-- -- - - ---_ ... .. -

A ( a t St • .)a",as)AND I (a t M I I a •

School). . ' ._ .- - .-- .' - ... _.- '- - .- . . - ----_. -.. -~-

W

f ~ \,2 __ ,. r" '-'--. --... --.--....... -----.---.- . _____ . ____ .E..-" :,.t r . f'. R

0. .. ~ .. r . c(

o o

o

a

o It'>

o

..

....

o .. H .... ....

01 AGRAM TO SCALE SHOWING ADJUSTMENT OF ALL

POINTS BETWEEN A AND I. (See dlaJ~am on paJe )

o • 0 0 1 I)

0.0010

0.0005

0.0000

Tne adjustment 01 these Intermediate Sanch Mark. can be effected by .cal in 3 the adju.t.ant correction. from the above dlaJralll,and subtractln3 aacn corraction from tne Ob •• ,.ved Elevation a. it appears In the Level Not e s ; 0 r, I f . des ire d , t h. a d j u 8 t m • n too r r e c t Ion sea n b e

•

•

•

computed by the fo I 10winJ formu la for tni ••• ctlon(1 to 1)1

FORMULA: Olst.A to Intar.Pt. Adjustm 'ent to Int.rmediata Point

z ____________ ~ _____ _

6. 25

I 'n" ]t :n I sea •• , ltth. A d J u • till. n t Cor rae t ion 8 are t 0 b a subt"acted from the Observed Elevatlons,as stated abov ••

MERAMEC SPRINGS AND REO BIRO OUADRANGLES. - -

PHELPS COU NTY. ST. JAMES AND DAWSON TOWNSHIPS.

The elevations in the followin~ list are b~sed on a precise-level line run in 1908 by E.L.McNair from the United States Army En~ineers'bench mark at Labadie Mo., the accepted elevation of which i s 472.572 feet.

The levelin~ under adjustment was run in the Fall of 1921 by the Vocational Department of the Missouri School of Mines and Metallurgy from the Dntted Btates ~olof!ical Survey bench maiDks at St. James and Knobview, the accepted el~va tions 0 f which are 1087.800 feet and 1074.005 feet respectfvely . .

-25-

MERAMEC SPRINGS QUAD~ANQLE.

( L at I t u d e '5 7· " 5 ' - '5 8 0; I 0 II :I i t u de 9 1 • '5 0 ' - 9 1 • " 5 ' )

(St Ja ..... Town .... lp)

LI NE FROM ST.JAMES NORTHWEST VIA JEFFERSON ROAD TO INTERSECTION WITH REPUBLIC ROAD, THENCE NORTH VIA REPU8LIC R 0 A D TOP H E L P S -G .A S CON 'A DEC 0 U N T Y LIN E •

Elevation Adjuatlll.nt Adjust.d No. D"cI"lptloll By F I ., d Book COl",.ectloll EI .... atloll

A. St •. Jam~s, corn~r of feet fee t J e et

M~r am~c ani 'Nashin~ton St,'s. , abo u t 70 ft. north of cen. t~ 'r lin~ of Frisco R. R. track, 25 ft. from do·~,neT of Comm~:~ci al Hot~l,in an ,~n ~ betw~en cross walk and siie walk; iron post stamp~d "1098". (s •• Bull.tln 1568 p a :I • 10. ) 1087.800 0.000 1087.800

RED BIRD QUADRANGLE.

L. , ( L a tit u d. '58· - '5 8 • 1 5 'j lon ' :I I t u d. 9 1 • '50 ' - 9 1 • " 5 ' )

St.James,0.5 miles north of,on Jefferson ~O~ at intersection of Parker Lane road, concrete culvert, north end wall,top of; :chistiled. squaTe.

K. st. James,!. 75 miles north of,on Jefferson road, at intersection of Republic road,on south side of road, 10" black jack tree, in base of;copper nail and washer ,

1078. 4'27

109:2.694

..;26-

0.000 1078.4''27

- 0.001 1092.693

P. B. M.

No. O.sorlptlon

1. st. James, 2. '25 miles north of, oJ: Republic road,Jefferson road,0.5 rui I es no rth 0 f, cuI vert, wes tend, sandstone slab, top of;painted

EI.Yatlon CO", By Fld.Bk.

fee it, 'fi e e t

Ad;" E I • 'y •

feet

square. ------------------------~ 1034.345 -0.001 1034.344

2. Jefferson road,l.O mile north of on 3epublic road,T road east,at Cox's residence,100 ft. east of, sandstone boulder in N. E.Cor. o f T road; topcorner brolen off,

painted white. ------------------ 1024.715 -0.001 1024.714

3. Jefferson road, 1.5 miles north of, Robinson creek,O.75 miles sout~ of, Fitzenreider's residence,lOO ft. west of,wa~on gate post,in base of; copper nail and washer. --------- 338.469 -0.001 998.468

4. Robinson Creek,0.25 miles south of, on Republic road,on east side of road, 10" post oak tree near Kenndy's ffiCiil box, in base of;nCiil. 9'23.575 -0.001 923.574

5. Jefferson road,2.25 miles north of, on Republic road, Robinson Cr~ek, west bank of,24" white oak tree, i~ root of;cop~er nail and washer. 917.604 -0.001 917.603

(Dawson Township)

6. Eo binson Creek,0. :25 miles north of, 15 ft. west of road, sandstone boulder on St. James-Dawson Twp. Line,oppdsite wire fence ~orner, point is on south hillside, 300 ft. south of Wooftchoppe~'s cabin; painted cross on top of boulder. 1022.334 .... 0.001 10.22.393

-..27-

E I • , a t Ion No. O •• cI"lp.tlon. By Fld.Bk. COl".

7. Robinson Gr1eek,0.75 miles nOTth feet feet

of,on Republic roarn,Township line,0.5 miles north of,at north end of rid~e, 30 ft. west of road, sandstone boulder, top of; chisel-

ed square. -------------------- 1039.423 -0.001

8. Robimson Creek, 1.0 mile north of, on Republic road,at foot o~ hill, concrete cuI ver t aero ss ro ad, W.

end wall, top of; chiseled Sg.--- 934.325 -0.00'2

9. Robinson CJr.eek, 1. 25 miles north of,Miles Sohool Souse, 1.0 mile south of,T road west,Denter of roads; road eldvation.--------- 976.005 ~0.002

10. Milds Bchool House,0.5 mil~sS. of,on Republic road,Marrs Cemetary,400 ~t.north of, east side of road,satidstone boulder at rail fence corner,top of; chisel-

Ad j d. E I • v. feet

1039.4'22

934.323

976.003

ed square. -------------------- 1059.077 -0.002 1059.075

11. 'St.James,6.0 miles north of,on Republic road,Sa:fe, ~ 2. 5 miles E.

of,Miles School, 25 ~t.W~of,S.W. Cor. S. E. c;tr.of the S. W. Qtr. Sec. 20,T.39N.,R.6w.,60 ft.S.E.of, concrete well curb, top of; bronze tab~et stamped" "

I. St.James,6.0 miles north of,on Republic road,·Safe,2.5 milesE. of,cross roads at Miles School, northeast corner of school li-oqse, li!I).estone wheel guard,top

-0.00'2

o f; painted s q~artl.-------------l061. 9'25 -0'.00'2 1061. 9'23

-2S-

ADJUSTMENT OF INTERMEDIATE TEMPORARY

BETWEEN THE POINTS1 Cat MI I •• Sohool) AND lie road on •• 11 •• outh of County Line.)

C'I •

0 ...

• • 2 o • c .... """1 LC"I

• I"-.. E .00 - ..

• 0 C'I

... -

.. It"

l"-. ... Ad j u. t. d

14

.. • .. "" 0 0 (\I "" 0

C'I N " f<'\

L I n • . 16 17

8 ENe H MARKS

PCon R.pub_

.. • 0 0 It" 0

I<' • 18 P 13 O"r

~~-------T------~------~------~--r-----~-------r----~ 1:.5

I

01 AGRAM TO SCALE SHOWING ADJUSTMENT OF ALL POI N T S BET WEE N 1 AND ( S e. d I a l r a '"

:: ' ."

t)

" P •

on p

Th. adjust",.nt of th ••• Int.r •• dlat. b.nch marks can be .ff.ct.d by

on

1.5""

<" ... -1

,.. .. ,-,

/ I'

scalln] th. adjust ... nt cor-

\,!':

--. f:~ . ,

r.ctions fro .. t .h. abo". dialralll,

< ' ...

and addln] .ach oorr.ctlon to th. ob •• ry.d .I.yatlon a. It app.ars In th. l.y.1 not •• ;or,lf d •• lr.d,t". adju.t •• nt corr.otlon. can b. co.put.d by th. followln] forllluia for thl •• ectlon (I to p).

FORMULA:

"" " ' R "< e' A (~ '+- 1

S E

",

........... " '-..

. ... 'C

(\I

0 . 0

• c 0 .. 0

• .. .. 0

(.)

.. c

• II

• " ... -<

0.028 x DI.t, I to Int.r.Pt. Adj. Cor. to In t.r. Pt. aC------- ___________________ _ -0.002)

4. 0

Add th ••• corr.ctlons to th. ob •• ry.d Yalu •••

El.Yatlon No. D •• crlptlon. By Fld.Bk. C,or.

106i~?~ 25 -6~662 1. (See desc ri ption on p ",ge )

12. MilesSchool,0.75 miles north of,on Republic road,W.L.Mart-in's h 0 us e, 150 ft. S. F. of, on top of hill,on east side of road, 12" post oak tree,in base of;copper nail and washer.--- 102~.857

13. Miles School, 1. '2 miles north of,on Republic road, Bourbeuse River,0.75 miles south of ford, at dim cross road,southeast corner of, sandstone boulder;

+0.003

painted sguare.--------______ 9'24. '251 :+0.006

.. 29-

Ad j d. E I • v •

1(~ll:j23

1023.8f)0

Elevation Ii 0 • D.scrlptlon • . By Fld.Bk. Cor.

feet feet

14. Miles School. 1.75 miles north of,on Republic road, Eourbeuse Ri ver, O. ':2 miles sou th of, farm house, 600 ft. east of,on east side of road, 15" walnut tree, in base of;copper nail . and washer. 873.567 :.0.010

15. ~ ash School House.0.75 miles south of,Bourbeuse River,at ford, 100 ft. south of,east bank of river, 30" sycamore tree, in base of; copper nail ani washer. 8~J. jj8 '+0.014

15a.Bourbeuse River, at ford,on Republic road, center of river; road elevation. --------.-:---- 834.000 :+0.014

16. Bourbeuse River, 0.25 miles N. of,on Republic road,at T.road east,northeast corner of,in base of fence post;copper nail

and washer. -----------------

17. Bourbeuse River,0.75 miles ~. of, Wash School, 400 ft. S.W.of, -J.W.Grossner's house, 300 ft. S.E. of,8 ft. west of three mail boxes,on west side of road,14" sealey bark hickory tree,in root

840. 106 :+0.016

of; copper nail and washer. --- 947.596 :+O.Olj

18. 'hash -School,O. 5 miles north of, on Republic road, Little Bourbeuse creek,at ford,onsouth bank,west side of road,lO" 8caleybark hickory tree,in base of; copper nail and washer. ------- 835.9'';.7 :+0.0'23

-30-

Ad j d. E I • v. feet

873. 577

840.012

834.014

840. ILL

947.615

835. j50

N ().

-P.

O,scl'lptlon. EI."atlon By Fld.Bk.

feet Wash School, 1.0 miles north of, Little Bourbeuse Creek, 0.5 Mi. north of,Harrison School,O.5 Mi. south of,Phelps-G9scon9de County Line,1.0 mile south of. Austin E. Breuer's hou se ,:200 ft. west of, at T.road east,west side of Rep-ublic road, ::2 ft. inside of fence, 10" post oak tree, in base of;

nail and washer. --------

COl' • feet

Ad j d. E I • " • feet

copper , __ .0.

I

. j : ~8 . 578 r---------~._~------~----------~~------~~

Adjust.d

j 28. 55' ~+0. o~6

.. ... 'C

C'I

0

0

..

+

c .. 0

c-... E .. .. • . .. " .. 0_ 0

,,0

c

C> C>

C>

.... '" C

o:: '.::

---

C>

C>

Lin ••

"" IX

o I

0·"

{" ...... -., .'Ii 4-

(,... ° 1

t oo. ~se I

r v "._ d ~· ! -I J.l --fl __ <: , .. _ ___ . __ J _ '" . '1') ~

c:'"l + .> c: +~

0.... ,- , ....

.. LinR _ .,J r'

<>

" t , I.. e

c ...

~.~ ~~j

--- .~ . (\.~ .

_ c- ...

2 ,-.

<> o

...

DIAGRAM TO SCALE SHOWING THE ADJUSTMENT OF ALL POINTS

BETWEEN P ANO G.

I t w I lb. 1' . ' '" ... b • I' • d t hat t h. I In. P toO I s a .pUl' lin. of 1.".1. In OUl' .ch .... of adjust •• nt. Th. lin. p to Q was I'ul\ as a pal'all.1 dupllcat. lin. of 1.".I',and the .o.t probabl. "alu •• fol' the .I.yatlon. of the point. on, u C h a I In .. I. t h. a". I' a 3. 0 f t h. ..". I' a I d. t. r. I nat lon, • Th. aao"e dla3ra ... how. a (red) In. of l.y.l. run by a Brandl. Inatru.erit and a (alu.) I In. of l.y.l. I'un by a GUl'I.y In.tru •• nt:a. th ••• In.tru •• "t. w.r. of equal r.fln_ •• " t, • qua I w • I .. h t • a. a p p I I • d to to tn. • 0 r k don. ' by. a c II In.tru •• "t. Th.(lr •• ") II"e I'.pr ••• "t. the ay.raJ' of tn. I " • t I' u .... " t a I d • t • I' • In.a t I 0 " • r In · t h e I. y. In. tad j u • till. " t ju.t pr.o.edlnl, It .a. found that the point P wa. 0.026 ft. low. A. the .pur lin. P to Q hlill.' directly on the point P,the" tne whole lin .. P to Q .1 I hay. to ae ral •• d 0.P26ft. I n tile d I a a I' a • t II I. a d j u • t •• "t I. ." 0 w " b Y t h. (a I a c k ) -1 In • .•

-3~-

Th. adjustlll.nt Is a constant of 0.026 ft. which Is to b. added to .ach of the ave"a~. inst,.u ... ntal obs.,."ed " a I u e. bet 10 •• n Pan dO. T his I. a p P •• r-. n t fro :~ the d I a ~ r a ••

FORMULA:

Adj.Cor.to Int.r.Pt.-Av.Obs Valu. + 0.026.

No. O •• cr-Iptlon.

-P. (See descri ption on paf1e 31 )

19. TNash School, 1. '25 miles north of, Little EourbeuseCreek,O.75 ~i. north of, HarrisonSchool,O. :25 miles South of,~helps-~asconade County 1ine,O.75 miles south of, Steve Lorts' house,70 ft.north of, T. road west(Mail road to Hif.lh jate in Maries Count7),no~thwest corner of roads, 'L4" oak stump in fence corner, top of; ten penny. nail.-----------------------Brandis

!20. Harrison School House, on St.

Gu rlev AveraE!e

James-Republic road,west side of School EuildinE!,north concrete door s~ep.top of; chisel-ed square. Brandis

"

Gurley AVeralte

,,-

'21. Harri son School House, O. 25 Mi. north of,on Republic road, - -Phelps-Gascoiade County Line, 0.'25 miles south of,pri'vate T.road west,southwest corner of,16" bl~zed black oak tree; bent nail in base of. ----- Brandis

Gurley AVera;1e

-32-

EI.vatlon Br Fld.Bk.

feet

j~8. 55' ~

96'2.732 96:2. 726

-96,"2:729

965.674 j65.658

-96'5:666

959.594 959.578

-9:59:586

Cor. feet

~o. G~I)

'+0.0'26

:+0.0'26

Ad j d. E I • v. fee t

9'28. 578

96'2.755

959. 61!~

No. Oe,crlptlon.

-

Elevation By FI d. Bk. Co r.

feet feet

Ad j d.

E I • v. feet

Q. Harrison School House,0.50 miles north of,on Republic road,intersectionSt.JamesRep~blic road and Hi~h GateRed Bird road,0.5 miles so~th

- -of,Phelps-Gasconade County Line,'28 ft. south of,center of road, 30 ft. Wt::lst <?f, 5 ft. insidt::l wire fenct::l,in line with center of road north into -GasconadeCo~nty, dim T.rosd west,50 ft. south of;iron post stamped "997 ".-----Erandis '

Gurl ey 997.242 997.'237 997.240'+0.0:26

-997.266 P. B. M.

1hlS co~pletes the deSCrtptlon and adJustment oj the l~ne

of level ,s fro71l 'St.}ames north12est v~a Jefferson Road to In

tersectlon ~~th Republ~c Road. t~ence north v~aRehubl~cRoad

to Pnelps-Gasconade CountyLIne.

MERAMEC S PRINGS OUADRANGLE.

(Latitude 37°45'-3So;lon3itude 91°30'-91°45'

( S t.

LINE FROM ST. JAMES EAST VIA OZARK HIGHWAY TO PHELPS-

CRAWFORD COUNTY LINE,SAID POINT ONE WILE NORTHEAST O.F KNOB

VIEW RAILROAD DEPOT.

ADJUSTMENT OF INTERMEDI ATE BENCH MARKS ALONG THE ROUTE

A VIA BAND C TO D.(A is the P.B.M. at St.,)a .. e. and 0 ia the

T.B.M. at the nterllectlon of the Ozark HI]hway and the Pnelps-

crawford , County Line.)

BELOW IS 01 AGRAM TO SCALE SHOWING THE ADJUSTMENT OF ALL

NTERMEDI ATE POINTS ALONG THE OZARK HI GHWAY BETWEEN"A"AT ST.

JAMES ANO"D"AT THE COUNTY LINE NEAR KNOBVIEw.

T h a affacted Jram, and a I e vat Ion bookajor,

adjustment of the.e Intar ... di at. ban en by aeal inJ the adjustmant corrections .ubtractlnJ each correction from the aa It appear. In the level note. and f de.lred,the adjuat ... nt corrections

putad by the f ° I I 0"" I n J for II u 1 a for t h a a act i on

FORMULA:

A D ,.I • 0 F. I N T E R • PT.· - - - - - - - - - - - - - - - - - - - )( O. 0 Ii 3 • 6. 2 5

( See D I a J ram 0 n pal e

Elevations pre v lou a I y

on this .ect l on adju.ted,aa follow.

A B C D

• 1087.8000 1074.6580 1083.3 6 60 1098.8580

f t , " • •

.. ... o o o

,

A. 22

::I( •

o 0 o ..

o 0

I , -:;1" ,

B 2~ 25 27 28 Adjusted

• ••• •• • o o ..

-34-

o It>

if I

C Lin e. • • o It>

o C'< . .. ..

..

t L o W E R

",arka c.n b. from the dl.observed description can ba eolOl-A toO:

/1T / fT : I

" I I I ,' IT,' · I ! I ,

I , I: : i :! I i I i I I i I 11 1

: I : I , : :! 10-I ' I. I ~ Ie I ~ I • 1, . I'

il : I I I I

I I I I

31 0

• It> .... It>

I.I ... I 11('\

i", 10 I • 1-0 r

I

No. D.scrlptlon. EI.watlon By FI d. Bk.

10~~~~00 A. (See description on pa ge 2 6 )

22. St.James,0.4 miles east of ton St. James-Cuba road(Ozark Hi~hway), Frisco R. R., 50 ft north of, tu rn in road to no rth; 500 ft. W. o~,concrete culvert across road, north end wall, top of; chiseled

square. ---------------------- 1065.363

RED BIRD QUADRANGLE.

Ad j- d. Cor. EI.".

0!666 1081~~60 P.B.M.

-0.003 1065.360

(Latl tude 38-~~8-15'; IO~3Itu~, 91-30'-91-'5') Qt J 1 0 °1 i.) ~. Ja~es .L O wnS .. l p ) B. u • ames, . m 1 e , . 1'., . 0.,

'''3 L. •

on Ozark giQhway,T.ro ad east (intersection of Ozark Hi~hway and Red Bird road),400 ft.S.of, Cr. F.A.Scott's house,300 ft w. of, on W. side of road, 10" elm tree,in root of;copper nail

and washer. ------------------ 1074.666

st. James, 1. ,5 miles N. E. o f, on Ozark Hi~hway,0.5 miles east of Red Bird road intersection,on N.side of road, 12" post oak tree,in base of; cop per nail

and washer. ----------------- 1062.669

'24. St. James, 1. 75 miles northeast of,on Ozark Highway, concrete culvert across road,south end

''"'5 L., •

wall, top of;chiseled trian~le. 1034.437

St.James, '2.0 miles N.E.of,on Ozark RighwaY,on top of hill, center of road; road elevation. 1054.446

-35-

-0.009 1074. 658

-0.013 1062.656

-0.015 1034.422

-0.017 1054.4'29

No. O •• crlptlon.

26. St.James, 2.5 miles northeast of,on Ozark Hi~hway, FriendShip School, 1.5 miles west of,concrete culvert(errected 8-13-20 by T.Hall),north end wall,toD

E I • vat I 0 'n: By FI d. Bk.

feet Cor. feet

of; chiseled square. --------- 1010.057 -0.021

'27. Friendship School, 1.25 miles west of,on Ozark HighwaY,at T. road N.to Hinkle P.O.,Corru~ated iron culvert,rock on east end of culvert,top of;chiseled

triangle. ------------------- 1018.92~ -0.023

'28. Fri~dshi p School, O. 5 miles W. of,on Ozark Hi~hway,Gerber's Store,600 feet W.of,concrete culvert,north end wall, top of; chiseled trian~le. ---------- 1062.045 -0.030

C. Friendshio School,'on Ozark Highway, southwest corner of concrete porch of school house, top of;chiseled triangle, oainted "1083.3"------------- 1083.400 -0.034

'C 1. F r I • n If • II IpS 0 1\ 0 0 I ,on 0 z a ,. k

Highway, 50 ft.S.W. of,in School Ylird,10 ft. from fence corner, 30 ft.N.of center of Ozark Highway, 30 ft.E.of center of T. road north;iron post stamped

"1083". --------------------- 1083.458 -0.034

29. FriendshipSchool,0.25 miles E. of,private T.road N.to Louis ~all's residence,75 ft.W.of, concrete culvert,S.end wall, top of;chiseled square. ----- 1037.041 -0~036

-36-

Ad rd. E I • v. fee t

1010.036

1018. )03

1062.015

1083.366

1083.424 P.B.M.

1037.005

No. O.scI"lptlon.

~O. Friendship School,O.9 miles E. of, on Ozark HiQhway,PhelpsCrawford County Lin~,1.1 miles west of,concrete culv~rt,south end wall, top of;chiseled

EleYatlon 8y Fld.8k.

feet COl" • feet

Ad j d. E I • Y •

feet

square. -,--------------------- 1042.297 -0.044 104'2.'253

31. J riendship School, 1. 5 miles east of,on Ozark HighwaY,PhelpsCrawford County Line,0.5 miles west of,Knobview,O.5 mi les north of across field, concrete culvert under Ozark Highway,north end wall, top of;chiseled square.- 1028.412 -0.049 1028.363

ID. Friendship School, 2.0 miles east of,on Ozark Hi~hway,at cross roads on Phelps-Crawford County Line,southwest corner of roads, 14" hickory tr~e,in base of; copper nail and washer. ------------- 10J8.911 -0.053 10)8.858

Ti/.1,'S completes the descrlpt-:-otn and adJustment of the

~'/,ne of levels from St. James east V1.a Ozark ll,.gi~1tay to the

P;~elps- 'Crau:jord 'County Luu .• sa,.d 1>olr~t o .. e rule northeast oj

Knobv~e1t Ra1.lroad ·£e/>ot.

P. A. R T . 1 I I .

THE LEAST SQUARE ADJUSTMENT OF A LEVEL NET

BY METHOD OF OBSERVATION EQUATIONS.

APPLIED TO THE ADJUSTMENT

OF THE

DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,

PHELPS COUNTY,

MISSOURI.

(BASED ON THe LINE OF LEVELS RUN BY THE U. S. GEOLOGICAL SURVEY IN 1908 ALONG THE FRISCO R. R. WITH PERM~NENT BENCHUARKS,cAw AT ST. JAMES, AND

WE- AT KNOBVIEW).

-38-

Showing ~lev8tions and closures when level circuits are arranged as shown in plat on page40.

It = 1087.800 - 25.875

(St.~ames ) Adjusted value.

I = 1061~925 (Miles School) -107.201

C = 1083.221 + 15.511

M = 954.724 (Royal) + 13.695

D = 1098.732 D ~ 1098.579 in arr.Cir.

J = 968.419 + 73.460

H = 1041.879 + 32.608

B =, 1074.487 + 8.734

C =, 1083 .. 221, - 82.223

F = 1000.998 - 80.066

G = 920.932 +177.647

D = 1098.579 - 24.844

+ C. J 53 Closure at C.

(Red Bird and Jakes Prairie Rds.)

(Red Bird Rd.and Ozark Highway.) H = 1041.879

- 40.44~ (Friendship ~chool ) _._--F = 1001.432 F = 1000.998 in arr.Cir.

+ 0 . 424 Cl osure at F. (Jakes Prairie and Tribby Rds.)

(Ozark Highway and County Line.)

E = 1073.735 (Knobview Depot.) E = 1074.005 (n ") Adjusted value • .

- C. 270

I = 1061.925 - 93.531

J = 968 . 394

Closure . At Vncbvi e~ .

J = 968.419 in above arranged circuit. - 0 . 025 Closure on circuit above . At

M = 954.724 - 33.201

G = 921.523" G = , 920.932 in above arra~gedcireuit.

Note: The values given on this page are not adjusted values, but only an arrangement to

f~cilitate the ad-- justmeat that i~ .J •

to be made on the following pages by

the theory ef Least Sqaares.

+ 0. 5"91 Closure on circuit above.At G. A =, 1087.800

- 13. 134 B =1074.,668

L B = 1@74 ... 487 ia sieve arraagedeire1li:t. + 0.179 Closure on circuit above.At B.

-39-

910 01 919 35' 9130' , 38°10' __ ~ __________ +-________________ ~ ____________ ~~ __________ -T~3~8°10'

- "*

38°00'

I G J S CON J DEC 0 U N T Y I (;, 0:- :Ii C\l

'r-- ---~-t .

I "':N' \

o

o

;;:ii

I

I I

I , I o .,.,

-, TT z, Jf2. • .;

<Pi \ •

~i \; ,-\

MARIES CO. V, ~~': --- -----1-- -;Q7-

I '0 , , P-i I !. .

:z; I I

H , '"

o ' .... C) p:: -'

III

....:.l If.J ~

~

:2:i ....:.l

;l!C

H 0

z. I

D cr \Q

S

• • ... , . ..... -+ -• • .., ... o c.. ... .

H'I :Iii •

o

o

o

Rd. X Q

) !

0:;

,.. 0 p ....

IQ

... r:. ...

.... +

;3:

<ll

~~o:; I ,," 0.' . ~

, '1-" 0 , no B v,ew

B. II. UT". us w. 8. ,It- 8.

:1 Q E-< .-. : 0"0" ___ JL __ ~~ __ ~~ __ ~ __ ~~~~~~~~ __ ~~J-~~R~.;;d~8~I ;r~d!· ~o;;u~a~d~r~.~n~g~4~.1·r-----~-----t-'~ ~

a.6 •• U .ra ~ ec Sp rin ge

91 401

10$7.880 8. S.

W N 91

-40-

S 35'

Ouadra nll i.

B I p

Observation Equations.

(A to I) +\11. .......... e , .... -. ....... -....................... = +0.000 (weight 0.160) (I to M) ".-... +V2 ...... -............................ -• .•• = +0.000 (weight 0.156) (M to J) · .... -..... + ~ ... ' .. " .............. " ................... = +0.000 (weight 0.312) (J to B) · ................. + ~ ...... ' ............... ' ......... = +0.000 (weight 0.625) (H to B) ........................ - ...... +. 'Vs .... ' ........................ = +0.000 (weight 0.294) (B to C) ....... ~ .........•. +Vs ............ ' ..... = +0.000 (weight 0.333) (C to F) •.•....•........•.• ~ ... +~ .............. = +0.000 (weight 0.333)

~~ to g~ .... ". " ................... -. ' ....................... +~ ..... = +0.000 ~weight 2.000~ to ......... -.... ................. ' ...... +V9= +0.000 weight 0.222 ~D to J~ -V1 -V2 -Va -V4 -Vs -ik -\; -i;{, -~ = -0.270 f weigh t 1. 000 ? ' I to · .. +V2 + ~ " ............................... -......... .. =, -0.025 . weight 0.625 (A to B) +V1 +V2 +Vs +v;.. +V5 ........... -.. ' ... . = +0.179 (weight 1.000) (H to F) ........ -..................... +Vs +~ +\; .. ............ = +0.434 (weight 0.5'71) (M to G)" ........ .. '. : +~ +V4 +Vs +'" +\; +~ = +0.591 (weight 0.124) (C to . D) ........ ' ... .................... . <It . . ....... +~ +~ +~ = ,+0.153 (weight 0 .. 445)

.. Formati.on of the Normal Equations •

Normal Equations are formed "in the following manner -from the Observation Equations:

RlJLE: 10 form the no~malequat, ,,o1ls 101" e'ack one of the un

known .quant: ~t .'Jes, 7Ilult'J.ply 'each o/}'se1"vat'f.on ' equat·'!.on by th~.p1"od-

9f ,tbewe$ght of ,that o~ser'Va~$onand ,the alge~~~$c coetf;c~ent

of that unknown quant'lity .Ut ·that ·equat:.on, and add ·theresfllt,s.

Having formed the several normal equations, 'their solution as simple simultaneous equations gives the most probable values of the 'unknown quantities •

•• • f.pf.~.l.u.ta •• ou •• quatro~. of tho fl~.t d.g~ •• ,tb. nor.al .~uatfon •• ay b •• olv.d by any of tho aI9.b~alo •• thod. ~f

.11.at1oft, by addltlon or .ubt~aotlo., by .ub.tltutlo., or by ooaparl.on. .n fact,th ••••• tbod. a~ •• att.faotory wh •• tb.~. ar. but two sq.ati0 •• to be .o~v.d. But In larg.r •• t., of thr •• or .or., 4t t. po •• lb4. t •• bort.n tho oe •• rloal rerk by tetrAg ad

••• ta •• of the p.ou~Jar .y ••• try whtoh a11 nor •• t .4eatto. 'o ••••• ~ .t I •• eob ••• t.r to .01 ••••• t of .or.a~ .4.atlo •• tha. a •• t of .rd'.arr,., •• fta •••••• quat.on. of tho .a ••••• ber whto' do aot

ha •• tb' •• r ••• try.

-41-

il Ill!

I\.)

,

I

Nor

mal

E

quat

ion

in V

1 •

.A"

.olu

t.

rer8

+0

.1S0

Vs.

•. •

.•

. .

· ..

•. '

.'

. .

• .

. •

'.'

. ..

· .

'.'

· ·

'.'

. ..

= +o

.uoo

oao

+1

.00

0V 1

+1

.000

V2

+1

.00

0V a

+1

.00

0V 4

+

1.0

00

V 15

+1

.00

0V d

+1.

000\

;-+1.000~ +

1.00

0VI'I

= +

0.2

70

00

0

+1

.06

0V 1

+1.000~ +1.000~ +1.000~ +

1.0

00

Vs·

••

.•.•.••••

•.•

• •

••

= +

0.1

79

00

0

1)

+2

. 160

V 1 +

2.0

00

\; +2.000~ +

2.0

00

'4 +

2.00

0Vs

+1

.00

0'k

+1.

000\

1; +

1.0

0b

'b +1.

000~

= +0

.44

90

00

Nor

mal

E

quat

ion

in ~ •

'.

'.

'0

• +0

.156

Vt:

·· •

..

. •

'.'

'.'

. .

• '.

' '.

' .

. .

. •

. .

. . .•

= +

0.00

00 CO

+

1.0

00

V 1 +

1.0

00

V 2 +

1.00

0'\k

+1.

0001

4 +

1.0

00

V 15

+l.O

OO

Vd +1.000~ +

1.00

0\18

+1.

000v

g =

+

0.2

70

00

0

• •

• •

+0

. 62

5V 2

+

0. 6

25V a

' •

• •

• •

•.•

•.•.

•

• •.

•.•.

• •

• •

• •

••

= -

00

01

56

25

+1

.000

V1

f,1..6

!D0'

k +

1.0

00

V 8 +

1.0

00

V 4 +1.000~·

• •

• •

• •

••

• •

• •

••

=

+0

.17

90

00

-

--_

. 2

) +2

.000

V1

+2

.78

1V 2

+2.625~ +

2.00

014

+2.0

00\1

5 +1

.000

Ve

+1.0

00v:

, +1.000~ +

1.0

00

\b =

+0.4

33

37

5

Nor

mal

E

quat

ion

in ~ •

.. '

.•

. '.

' .

. '.

+0

. 312~'

. .

. .

• •.

.. .

. .

. .

• .

. .

. .

. .

. ..

=+

0.00

0000

0 +

1.0

00

V 1 +

1.0

00

V 2 +

1.00

0Va

+1.

000V

4 +

1.0

00

V 15 +1

.000

Ve

+1.0

00v:

, +1

.000

Ve

+1

.00

0Vg

=

+

0.2'

1000

0 •.

• •

• +

0. 6

25V 2

+

0. 625~·

• •

','

• • .

• ,

•.•

• '.

' •

• •.

• •

• •

• '

.'

=. -

0.0

15

62

5

+1.

000V

t +

1.0

00

V 2 +1.000~ +

1.00

0V4 +1.000\~ •

.. •

• •

• •

• •

• •

• •

••

=

+0

.17

90

00

•

•• '.

' •

0.'

+0

. 124

V8 +

0.12

4V4

+0

. 124

V 6 +O

.124

Vd

+0

. 124

v:, +0.124~

• =

+0

.07

32

84

0

3)

+2 .O

OO

V 1 +2.825~ +

3.06

1Va

+2 •

124

i4 +

2. 1

24V 1

5 +

1. 1

24V 8

+

1.1

24

\; +

1.12

4Ve

+1

.00

0"

=,+

0.5

06

65

90

Nor

mal

E

quat

ion

in V

4 •

· ••

. ' .

•.•

'.'

. •

' .•

+0.

625V

4•

• •

• •

• •

eo

. •

••• '

• •

• •

• '

. =

+0

.00

00'

00

+1

.00

0V 1

+1

.000

~ +1

.000

Va

+1.0

00\1

4 +

1.00

0V15

+1

.00

0Vg

+

1.0

00

'h +1.

000~

+1

.00

0Vg

:;;

+0

.27

00

00

+

1.0

00

V 1 +

1.0

00

V 2 +

1.0

00

V 8 +

1.0

00

'k +

1.00

0V15

'·

• •

• •

••

• •

• •

• •

••

= +

0.1

79

00

0

" •• •••

.••

. +0

• 12

4Va

+0 •

124

V 4

+0 .124~ +0

• 1

24Ve

+0.

124\

1, +

0.1

24

'k·

•.

=

+0

.07

32

84

4) +

2.00

0V1

+2.0

00V

2 +

2. 1

24V8

+2

.74

9'4

+2 ~ 1

24Vs

,!"1

.124

V8

+1.

124¥

, +1

.124

~ +

1. OOO

~ =

+

0.5

22

28

4

~rmal

Equ

atio

n in

Va

• A

b,o

lut,

Tlr

. •.

~

.'...

....

. .

. ..

..

.. •

•.

. .

+0

.29

4 V

a·

. ..

...

. .

...

. .

. . ..

= +

0.0

00

00

0

+1

.00

0V 1

+

1.0

00

V 2 +

1.00

0Vs

+1.0

0011

4 +

1.0

00

V 15

+1

.00

0V 8

+1

.000

1.; +1.000~ +

1.0

00

i.b::

+

0.2

70

00

0

+ 1

. 000

V 1 +

1. 0

00V 2

+

1. 0

00

'k +

1. O

OOi4.

+ 1

. OO

OVS

' •

• •

• •

• •

'.'

• • •••

:: +

0.1

79

00

0

' .

• 0

.••

••••

••

'.

' •••••

+0

. 571

V 5 +0

.571

V8

+0

. 571

\Z,

• •

• .

• ••

:: +

0.2

47

81

4

• ••

" .•••

+0

. 12

4'k

+0

.12

4'4

+0

. 124

Vr;

+0.1

24V

8 +0

.124

\1, +0.124~· ••

:: +

0.0

73

28

4

5)

+2.

000V

1 +2

.000

V2

+2

. 124

Vs

+2.1

2414

+2

. 989

VSti+

1. 6

9%V e

+1.

695\

1, +

1. 1

24Vs

+1

.00

ml,

=

+0

.77

00

98

Nor

mal

E

quat

ion

in ~

• •

· •

. .

• .

. .

~.

..

' .

. .

. .

. .

. +

0. 3

33V g

' •

• •

• •

• •

• ••

= +

0.0

00

00

0

+1.

0oo

Vt

+1

.00

0V 2

+1

.000

Vs

+1.0

0014

+1.

000V

s +1

.000

Ve +1.00m~ +1

.000

Va

+1.

00m

b =

+

0.2

70

00

0

• •••••••••••

0 ••• +0.571~ +

0.5

71

, +

0.5

71

\I,·

•

• •

• =

+0

.24

78

14

• •••••••

+0

. 12

4'k

+0.

124i

4. +

0.12

4VS

+0

. 124

Ve +0.1~4\z' +

0.12

4Va

• =

+0

.07

32

84

----

. !.

6)

+1.0

00V

1 +

1.0

00

V 2 +

1. 1

24V

s +1

.124

114

+1

.69

5V I

S +

2.0

28

Ve

+1.6

951.

; +

1. 1

24Vs

+1.

00m

b =

+0

.59

10

98

\.

.)J I

Nor

mal

E

quat

ion

in \

Z,

•

· '. .

. . '.

' +0

.333

\Z,

• =

+0

.00

00

00

+

1.0

00

V 1 +

1.0

00

V 2 +1

.000

Vs

+1

.00

0'4

+1

.00

0'k

+1

.00

0V 8

+

1.00

0\z,

+1.

000V

a +1

.00C

J\t,

=

+0

.27

00

00

•

• ••••

.•••••••••• +0.571~ +0

.571

Ve

+0

.57

1\Z

,·

• •

• •

••

= +

0.2

47

81

4

• ••••

• ••

+0

. 124

1k +

0. 1

24i4

. +

0. 1

24V 5

+

0. 1

24V6

+0.

124\

1, +

0.1

24

\b·

••

= +

0.0

73

28

4

•. '

. •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• +

0. 4

45ib

+0

. 445

Va +

0. 4

45,b

==

+0

.06

80

85

7)

+1

.00

0V 1

+1

.00

0V 2

+

1. 1

24V 8

+

1. 1

24V 4

+1

.695

\+;·

+1

. 695

V 8 +

2.47

3V,

+1

. 569~ +

1. 4

45.b

=

+0

.65

91

83

Nor

mal

E

quat

ion

in V

a •

• •••••••••••••••••••••

• •••••

+2

.00

0,·

••

=

+0

.00

00

00

+

1.0

00

V 1 +

1.00

01b

+1

.00

0\h

+1.

0001

4 +

1.0

00

V 15

+1

. 000

V 8 +

1.0

00

v' +

1.00

01,t,

+1.000~=

+0

.27

00

00

•

•••

0 •••

+0

. 124

V 8 +

0.1

24

'4 +O.124~ +

0.12

4V6 +0.124~ +

0.1

24

,· ••

=

+0

.07

32

84

•

• •••

• •

• •

• •

• •

• •

• •

• •

• •

• •

• +0

.445

iJ,

+0

.44

5, +0.445~=

+0

.06

80

85

8)

+1

.00

0V 1

+1

.000

'12 +1.124~ +

1. 1

24V4

+1.

124V

e +1

.124

Vs

+1

. 569~ +

3.5

69

' +

1. 4

45Vg

=

+0

.41

13

69

. I ~ t

Nor

mal

E

quat

ion

in ,

•

• . .•

.,

•

'. • _

. •

• •

• •

• •

• •

• •

• •

• •

• ..

• •

• •

• •

• •

+0.

222"

", =

+0.

0000

00

+1.

000V

1 +

1.0

00

\; +

1.00

0Vs

+1.0

0014

+1.000~ +

1.00

0Vs

+1.0

00\;0

+1.000~ +

1.00

0"",

= +

0.2

70

00

0

• ••

• •

••

• •

• •

• •

• ••

• •

• •

• •

• •

+0

.44

5\ +O~445' +

0.4

45

' =

+0~O

S808

5 9

) +1

.QO~V1 +

1.00

0V2

+1.0

ooV

a +1.000~ +

1.00

0Ve

+1.0

00V

s +1.445~ +

1.4

45

' +1

.667

"", =

+0.

3380

85

1)

2)

3)

4) 5) e'~

7)

8)

9)

( 0

)

NO

R M

AL

ill

QU

A T

IO

N S

•

V 1

~

~

~

~

~

.~

~

~

Ab

eo

lute

T

erM

-+

2.1

60

+

2.0

00

+

2.0

00

+

2.0

00

+

2.0

00

+

1.0

00

+

1.0

00

+

1.0

00

+

1.0

00

-0

.44

90

00

= 0

+~.~

~~ ~.781

+2

.62

5

+2

.00

0

+2

.00

0

+1

.00

0

+1.

000

+1

.00

0

+1

.00

0

-0.4

33

37

5 =

0

+S.~

00 +~~g~~3.061

+2

.12

4

+2

.12

4

+1

.12

4

+1

.12

4

+1

.12

4

+1

.00

0

-0.5

06

65

9 =

0 +~

.~0~

+~

.~~~

+2

.~4~

~49_

+2

.12

4

+1

.12

4

+1

.12

4

+1

.12

4

+1

.00

0

-0.5

22

28

4 =

0

+~.~0,~

+!

.X10

?J

+i.J

~~

+~.i

l4 L

:2.9

89

+

1.6

95

+

1.69

5 +

1.1

24

+

1.0

00

-0

.77

0'0

98

=

0

+1.0

00

+1.~0e

J +1.12'~

+t.

1'z,4

' +r~;

5t2.

028

+1

.69

5

+1

.12

4

+1

.00

0

-0.5

91

09

8

= 0

+1.E'J

~0 +!

.00e

J +

l.!l

,4

+1.

12'4

+1

.8>5

3 +!

~gg3 ~:

~~~ +

1.5

69

+

1.4

45

-0

.65

91

83

=,

0

+1.0

eJeJ

+1

.0€'>

£1

+l.!l~

+1.1

:24

+l.

!2A

+,

3:.1

24

+1

.56

9 I +

3~69

+1

.44

5

-0.4

11

36

9 =

0

+.ir.00

'~ +1.~0

+1.k.

10t6

+1

.k.1k

.10'

+1.

00'0

+

1.0

00

+~.

4M.5

+1.,(~1.667

-0.3

38

08

5 =

0

(Nu

.ber.

b

elo

w

the

red

li

n.

are

re

du

nd

an

t n

u.b

era.)

I ~

(p)

Sol

utio

n of

Nor

mal

E

quat

ions

.

Term

1

2 ;::

fl.

-,.i

()

7 ,-- oj

~ ....

bso

lute

'P!.

+2 .16~

+ 2.

00

00

0 +

2.0

00

00

+

2.0

00

00

+

2 .0

0000

+ 1.0

00

00

+ 1

.00

00

0

+ 1

.00

00

0 +

1.0

00

00

-0

.44

90

00

P1

P2

(-o

.46

30

) -0

.92

60

0 -

0.9

26

00

-0

.92

60

0 -

0.9

26

00

-0

.46

30

0 -

0.4

63

00

-0

.46

30

0 -

0.4

63

00

+

0.20

7887

&

P e

P ..

PI

PI

• +

0.92

900

+0.

7730

0 +0

.14B

OO

+0

.14B

OO

+

0.07

400

+0.

0740

0 +

0.07

4,00

+O~07400

-0.0

17

60

1 f

l! (-

1.0

76

4)

-0.8

32

06

-0

.15

93

1 -

0.1

59

31

-0

.07

96

5 -

0.0

79

65

-0

.07

96

5 -

0.0

79

65

+

0.01

8946

a

. -..

• ••••

+0.

5658

2 +

0.14

886

+0.

1488

6 +

0.13

643

+0.

1364

3 +

0.13

643

+0.

0124

3 -0

.07

62

40

&

(-1

.76

73

) '-{

).26

30B

'-0

.26

30

8 '-

0.2

41

11

-0

.24

11

1 -

0.2

41

11

-0

.02

10

7

+0.

1347

39

P e

Pr ••••

• •

• •

+0.

8342

6 +

0.20

926

+0.

1503

2 +

0.15

032

+0.

1503

2 +

0.05

894

-0.0

83

64

9 :s

Pe

•••••••••••••

(-1

.19

87

) -0

.25

08

4 -

0.1

80

19

-0

.18

01

9 -

0.1

80

19

-0

.06

06

5

+0.

1002

70 -~

'f

P,

••••••••••••••••••

+1.

0217

7 +

0.68

361

+0.

1126

1 +

0.11

261

+0.

0441

6 -0

.31

04

80

~

P iO

•

• •

• •

• •

• •

• ••

(-0

.97

87

) -0

.66

90

5 -

0.6

69

05

-0

.11

02

1 -

0.0

43

22

+

0.30

3867

P 1

0

P u

P1-2

•

P u

P14

P u

P1

8

• •

P u

Pu

•

.-.

•.

••

+1.

0417

6 +

0.70

876

+0.

5197

9 +

0.48

794

-0.1

40

62

7

&1

.(-0

.95

99

) -0

.68

03

4 -

0.4

98

95

-0

.46

83

7

+0.

1349

88

&2

+1.

0045

6 +

0.61

116

+0.

6009

7 -0

.11

30

38

P1

S •

• (-

0.9

95

5)

-0.6

08

41

-0

.59

82

7

+0.

1125

29 P

1~

• •

• •

••

+2.

3965

3 +

0.34

852

+0.

0045

32 &

5 •

• ••

(-0

.41

72

7)-

0.1

45

43

-0

.00

18

91

P!.6

• •

+0.

5535

9'+

0.02

4204

&7

(-1

.80

64

) -0

.04

37

22

&8

, ~ 0'\

I

''(q

) S

olu

tio

n o

f N

orm

al

Eq

uati

on

s.

Ab

sol

ute

-~ t t

Te

rm

+

2.7

81

00

+

2.6

25

00

+

2.0

00

00

+

2.0

00

00

+

1.0

00

00

+

1.0

00

00

+1~00000

+1

.00

00

0

-0.4

33

37

' .1

.1,2

00

-1

.8,2

00

-1

.8'2

.00

-1

.8,2

00

-0

.92

60

0

-0.9

26

00

-0

.92

60

0

-0.9

26

00

+0

.41

57

74

2 4

6 8

C? 3

Ii 1

+0

.92

90

0

+i.

71

30

0

+o

.lia

oo

+

0.1

48

00

+

0.0

74

00

+

0.0

74

00

+

0.0

74

00

+0.01400

-~0~~17601

I • •

Cho

• q,

.~ •

EJt2

• E

!t.s.

Q

:u.

<ha

•

i.·~ .

. 8 '.,

~o.

~1

•

t:·

ctl4

."

qU

I.

t~ . I~: :

. 0

, 1

, ·2

, 8

'. E

lS4

' ~e

•

" • .. •

L(\

}se

. •

"b7

'iae.

~t .

. •

EJt.o

•

(!JJ

1

• ..

• •

•

•

• "

"

+3

.06

10

0

+2

71

24

00

-.2

71

2'i

OO

';I:

1'2

40

'O+

1:1

24

'Oo

+1.1240~l:OO'OOO:W06659

.1.8

52

00

-1

.8,2

00

-1

.8,2

00

-0

.92

60

0

-0.9

26

00

-0.9

26

00

-0

.92

60

0

+0

.41

57

74

-0

.64

31

8

-0.1

23

14

-0

.12

31

4

-0.0

61

57

-0

.06

1,7

-0

.06

1'7

-0

.Oi1

'7

+0

.01

46

45

+

0.,

6'8

2

+8

.14

88

6

+0

.14

88

6

+0

.13

64

3

+0

.13

64

3

+0

.13

64

3

+o

.oli

ii -0

.07

62

40

• -

. -. -

:---

+2"7

14 9

00+2

"7i 2

4 0

0

+1

. 'I i'

i'O'O

+i':ii

ioo

+i •

12 40

0--;"

'l.-:-o

'000

0'-:

0:-

52

22

84

•

• ••

-,J.

.8H

OO

-1

. H

20

0

-0.9

26

00

-0

.92

60

0

-0.9

26

00

-0

.92

60

0

+0

.41

57

74

.

0.0

23

,8

-0

.82

,,8

-0

.01

17

9

-0.0

11

79

-0

.01

17

9

-0.0

11

79

+

0.0

02

80

4

•

• • "

•

• " " • •

• •.

• • •

· • "

•

" •

.0.0

39

16

-0

.03

91

6

-0.0

35

89

-0

.03

58

9

-0.0

35

89

-~.00327

+0

.02

00

57

+

0..

83

42

6

+0

.20

92

6+

0.U

03

2

+O

.1!5

03

2

+0

.15

03

2

+0

.om

4

-0.0

83

64

9

--.-. -

. -. -

+"2

;989

0'0-

"+1;

695"

0"0

+1

. 6

95

0 0

+

1 .1

240

0--;"

1:00

00

0:0:

1"'1 0 0

9"8 '

-1

.85

20

0

-0.9

26

00

-0

.92

60

0

-0.9

26

00

-0

.92

60

0

+0

.41

57

74

'. • •

-0

.02

35

a-0

.01

17

9

-0.0

11

79

-0

.01

17

9

-0.0

11

79

+

0.0

02

80

4

-0

.03

91

6.0

.03

58

9

-0

.03

58

9-0

.03

,89

-0

.00

32

7

+0

.02

00

57

-0

.05

24

9

-0.0

37

71

-0

.03

77

1

-0.0

37

71

-0

.01

47

8

+0

.02

09

83

+

1.0

21

77

+

0.6

83

61

+

0.6

83

61

+

0.1

12

61

+

0:0

44

16

-0

.31

04

80

---------+2;02iO'O-;I.69,00-;t;12400~:OOOOo_:0:_i9lO9"8

.0.4

63

00

-0

.46

30

0

-0.4

63

00

-0

.46

30

0

+0

.20

78

87

•

-0.00~89

-0.0

0,8

9

.0.0

05

89

-0

.00

58

9

+0

.00

14

02

-0

.03

28

9·-

-0.0

32

89

-0

.0'3

28

9

-0.0

03

00

+0.018~82

-0.0

27

09

.0

.02

70

9

-0.0

27

09

-0.0~062

+0

.01

50

73

.0.457~7

.0.4

'75

7

.0.0

75

34

-0

.02

95

5

+0

.20

77

27

¥i

~ill.a

+0

.70

87

6 +0~51979

+0

.48

79

4

-0.1

40

62

7

--·-.;-7---~.47300-~56900~

l:44iOO-:O:6591B~

-0.4

63

00

-0

.46

30

0

-0.4

63

00

+

0.2

07

88

7

-0.0

05

89

-0

.00

58

9

-0.0

05

89

+0.O~1402

-0.0

32

89

.0

.05

28

9

-0.0

03

00

+

•• 0

18

38

2

-0.0

21

09

-0

.02

70

9

-0.0

10

62

+0~015073

-0.4

51

37

-0

.07

'34

-0

.02

95

5

+0

.20

77

27

•

• ••

.0.4

82

20

-0

.3'3

65

~0.33197

+0

.09

56

74

+

1.0

04

56

+0.61116+0.600~_O.113038

+3

.,6

90

0

+1

.44

§0

0

-0.4

11

36

9

• •

• •

••

-0.4

63

00

-0

.46

30

0

+0

.20

78

87

•

••

-0.0

05

89

-0

.00

58

9

+0

.00

14

02

-0

.03

28

9

-0.0

03

00

+

0.0

18

38

2

• ••

-0.0

27

00

-0

.01

06

2

+0

.01

50

73

-0

.01

24

1

-0.0

04

81

+

0.0

34

21

8

~O.25925

-0.2

43

46

+

0.0

70

16

6

-0~371a4 -0.~6s64

+0

!06

87

73

+2.396!)~

+0

.34

85

2

+0

.00

45

32

:;

}.6

67

00

-:O

:-3

38

08

5

-0.4

63

00

+

0.2

07

88

7

-0.0

05

89

+

0.0

01

40

2

-0.0

00

27

+

0.0

01

67

5

• -0

.00

35

7

+0

.00

50

73

q.,. 2

.

....

e.

. .•

• •

.0.0

01

91

+

0.0

13

41

9

-0.2

28

54

+

0.0

6'8

6§

-0

.3S

95

4

+0

.06

76

27

~$.

• 4

"

• '. •

• -0

.05

06

9

-0.0

00

65

9

40

.5§

3S

9

+0

.02

42

04

--

----

----

----

----

~ te

.~ ~ ~ <).0

~!

<Its

E!a

.4

<Ita

~~

~:

C}ao

~!

~3

~4

~5

C}2

e GJ

27

~e

'l?9

<60

qu

ce~

'1:13

te

4 '1:

Is ~~

<bs

q,g

q.,o

Q

l.1

~2

~!

" 11:1-~

'I

(_)

Sol~tion

of N

orm

al

Equ

atio

ns.

Vi

v<;>

v. .. V4

V R

v'"

V7

v'" lJ,

.tt, +0.2978$'7+0.0l8946

+0~13473g

+0.

1002

70 +0.303867+0.1~4988

+

0.11

2529

-0

.00

18

91

-0

.04

37

22

,rR

+

0.02

0243

+0

.00

34

82

+0.

0009

61 +

0.G

0265

2 +

0.00

1890

+0.

0204

78 +

0.02

6158

+0.0063~Ci

Fa

-0.0

02

06

8 -

0.0

00

35

6 -

0·.0

0107

7 -0

.00

08

05

-0

.00

04

92

-0

.00

22

29

-0

.00

27

18

+

0.0

04

46

7

.~ ~.062954

, -0

.01

08

30

-0

.03

27

83

-0

.02

45

00

-0

.09

09

70

-0

.09

25

05

+

0.13

5969

1'8

-\

).0

28

11

9 -

0.0

04

83

7 -

0.0

14

64

3 -

0.0

10

94

3 -

0.0

40

63

3.0

.06

07

32

~

·-o~160811

-0.0

27

66

6 -

0.0

45

68

7 -

0.0

43

56

1 +0.17~622

17

-0.0

21

40

3 -

0.0

03

68

2 -

0.0

06

08

1 +

0.02

31.1

3 C

heck

in

Nor

mal

E

quat

ion

(1):

,

(P

ag

_)

-0.0

64

01

5

+0.

0708

58

1h

-0.0

32

80

7 -

0.0

29

47

9

+0.

0354

29

-0.1

08

84

4

+0.

0462

26

'lb

+0.

0503

95 -

0.0

54

42

2

r10

-0

.02

96

37

L1N

I A

to

I

to

M

to

J to

H

to

B

to

e

to

F

to

G

te

Dte

O ••• '''I.d

fil

ff.,

.no

. I'

n .1

...

..

1=

-2

5.8

75

M

=

-ie

7.2

01

J

= +

1

3.6

95

H

=.

+73

.460

B

=

+ 3

2.6

08

C

=

+ 8

.73

4

F =,

-8

2.2

23

G

,=

-8

0.0

66

D

=

+

177.

647

E

= -

24

.84

4

,Aaj

ustli

re11

t Q

uaD

titi

est'.

-0.0

43

72

2

+0.

3473

24

V 1 =

-0

.02

96

VIS

=+

0.17

36

-0.4

49

00

0

+0.

060

~~2

~= -

0.0

54

4

~ =

+0.

0607

+0

.135

Qo~

Va =

+0.

0354

V,

=+

0.13

60

__

__

__

_

+0

.00

44

61

~=

+0

.02

31

~=+0.0045

-0.6

65

58

1

+0.

6655

76

~ =~.0~37

-0.6

65

58

+

0.6

65,

58

, A

dT

u •

t •

d

E x

o.

I I. n

t 0

h •

0 k

•

o 0

, r

eo

t I

0 1\

fA

~, t

n ~

oe

Ad

JU

S te

d

E leva

t ,. 0

11

0 f

.J un

c t

,. 0

n P

o t,

n t

· s.

:':'0

.029

6 ~ -

25

, .. S04

6 ft

. In

D

aw

son

TO

rl'n

sh

"p

,Ph

elp

s

Co

un

ty.

-0.0

54

4

=-1

07

.2,5

54

II

! ~

lo8~

8oo

fi'r

h =

1

04

1.8

55

5

ft.,

+

0.0

35

4

=~ 1

3.7

30

4

II

-8~.9046

+ 9

2.78~?

~

+0

0231

:;+

7

3.4

83

1

II

• ---

'1-,

+0:

1737

=+

3

2.7

81

7"

J =

10;

61',

695

4

/I

J:;

=

+0.

0607

=+

8

.79

47

"

-10

'7_

85

54

+

0.13

60

=-

82

.08

70

" k

=

9~4

.?~

04

/I

Ie

=

+0

.00

45

=~

~8e.

e6:\

.5

" +

1

9. ?~

\ -0

.04

37

=

+1

77

.60

33

" I

== 9

68

. C,

'!0

4

/I

f [t

=

-0.0

35

7,

=~

24

.87

97

"

+'7

3_

48

g1

10

'iL2

. 6 s

s:,~

11

'l

"~

+

8.

79

4?

,l

ti ~

~

«:I

~\j

\()

108c

,.4~99

11

• •

ti ~

-88

.08

?0

\~

......

-.

0-

11

" ...

l.O

Ol.

.S

4!<.

9

-8

0.0

62

.5

II .

L =

1041

.6F

,~r;

, "

IG =

9~

:Z. ~8

14

II

(.!)

:::

::;

l"

t--I 0

'I

' 0

-l{

) l>

;)

t--<:>

~

oo

c •

.. •

l>;)

"<

t 'l

" 0

'\ 5~

l'"

0 C

...

... II

" <-

:\ ~

I

PAR T I V

THE LEAST S~UARE AD.JUSTMENT OF A LEVEL NET BY METHOD OF CONDITION EQUATIONS

AND CORRELATIVES.


OF THE

DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,

PHELPS COUNTY,

MISSOURI.

(BASED ON THE LINE OF LEVELS RUN BY THE U. S. GEOLOGICAL SURVEY IN 1908 ALONG THE FRISCO R. R.

WITH PER.ANENT BEN~HUARKS wAw AT ST. ~AUES,

AND wE- AT KNOBVleW).

-48-

91°30' 38°10'--+-------------~------------------------------------______ _+----------------~8°10'

I

I GAS CON A DEC 0 U N T Y. ! T--J;£. -H E -~ P -:.~:~ (; - U N -T-;-. ~::- -.Ll

• f ij ~. ;; :1.

;>11 ,> I

" ,..... ""<", . ...1 I

i"i ';'1 I

D SON I p: I:

MAR I E S I

C 0 U N T Y. I I

--- - -- -~------I

D ILL 0 N I

:S

TOWNSHIP

...... ~ <\2 1

~ i-,

o

o

o

38°GO'~~------------~--~~~~~~~======~------~~~~----~-----------~8°eo· 91 40' ,-.~_ ............. l8t. James 91 0 30'

U. S.P .. B.Il. 1087.800

-49-

Least Square Adjustment of Level Net.

Link Observed From To Ditterenoe

A to B B" C C" 0 0" E C" }P

F! G D" G

- 13.134 + 8.734 + 15.511 - 24.'844 -82.223 - 80.086 -177.647

}P" H + 40.447 B" H - 32.608 H" J 73.460 A" I - 25.875 I" J - 93.531 J" W - 13.695 I" M -107.201 G if M" + 33. 201

Name

a b o d e t , g h i j k 1 m n p

Miles Distanoe

D.

1.0 3.0 2.25 1.0 3.0 0.5 4.5 1.75 3.4 1.6 6.25 1.6 3.21 6.4 8.05

P-.eight _.1_

D

1.000 0.333 0.444 1.000 (j.333 2 .. 000 0.222 0.571 0.294 0.625 0.160 0.625 0.312 0.156 0.12'4

Derived Correotion

-0.0306 +0.0808 -0.0569 -0.0354 +0.1387 +0.(1047 +0.0453 +0.0633 -0.1731 -0.0236 -0.0294 +0.0076 ·0.0353 '-0.0543 +0.1568

Con d i t ion E qua t ion s •

1) 0 = 0.062 - ' d - c b a 2) 0 = + 0.153 + g - f e + c 3) 0 = - 0.434 + e + h i + b

4) 0 = -0.157 h + f + p m - j

5) 0 = + 0.204 + a + i + j - 1 k 6) 0 = - 0.025 + m n + 1

correote'd Ditterenoe

- 13.1846 + 8.7948 + 15.4541 - 24.8794 - 82.0863 - 80.0813 -177.6017 + 40.5103 - 32.7811 - 73.4836 - 25.9044 - 93.5234 - 13.7303 -107.2553 + 33.3578

Each complete circuit furnishes the condition that the sum of its adjusted differences of elevation shall equal z&ro when given the proper signs as if run continuously around the circuit1clockwise or counter-clockwise.

Number of Conditions = (No.of lines) - (No. of Jnct.~oints) + 1. " " " " = + 15 10 + 1 = 6.

The minus signs applied to the small case letters in the above condition equations result from changing the direction of the arrows (when red arrow goes in opposite direction from the black arrows in in diagram on preceeding page) so as to be continuous around each circuit. It is not necessary that all the circuits be traversed in the same direction in a given problem1but forelearness the work will appear more cODsi8tant if all the eireuitsare traversed in one direetiOb,eitbereloekwise or counter-elockwise.

I 'JI ~

I

TABL

E OF

O

OR

REL

ATE

S.

Pro

du

ot

of

Os

wit

h Oorr~lates.

+.3

535

+.1

806

+.5~

63 +

.1~4

8 +.

O~71 +

.0~4

8 L

ines

~p

1 -1

a b o d e r g h i j k 1 11

1 n p

.10

.3

0

.22

5

.10

.3

0

.05

-1

-1.

.. 1 • • · . •

• .4

5

.17

5

••

.84

• •

• •

.18

.8

25

••

• •

.18

.3

2'1

••

••

.84

.8

05

••

2 • •

• •

+.1

• •

-1

-1

+1 • •

. . . · . • •

• • • •

• •

• •

3 4

5 +

1

6 ~ o o o

Sum

(Sum

x ~p)Lines

• •

+1

• • • •

+1 • •

• •

+1 ~1

• •

• •

; .. • •

• • • •

• • · . • •

• • • •

+1

• •

-1

• •

-1

• • • •

-1

• •

+1

• •

• •

• •

• • • •

• •

+1

+

1 -1

-:-

1

• • • • • •

• • · . •

• •

• • •

• • • •

• •

• •

• •

• •

+1

+1

-1

· .

-1 o o

+1

o o o -1

o o -1

+

1

-.3

53

5

• • •

• ••

• •••

• • •

•••

+.0

47

1

• ••

••

-.3

53

5 .•

••••.•

+.5

56

3 ....

.....

• •••

• • •

•••

-.3

53

5

t,1

00

6

•••••

• •••

• • ••

••

• ••

••

-.3

53

5

• • •

• ••

• •

•••

• •••

• • ••

••

......

• •••

••

-.1

00

8

+.5

56

3

• ••

••

• ••

••

• ••

••

• ••

••

-.1

00

6

•••••

+.1

94

8

• ••

••

• •••

• • •

•••

+.

10

06

•••••

• ••

••

• ••

••

." ...

. • ••

••

• ••

••

· ....

· .. " ..

+.5

56

3

-.1

94

8

-.5

56

3

•••••

• ••

••

• •••

• +

.04

71

• ••

••

• ••

••

'0' ••

• • ••

••

. • II ••

.... . . .

• ••

••

• •••

• • •

•••

• •••

• • •

•••

· ....

• ••

••

• •• I

.'

• •••

• • •

•••

• ••

••

• •••

• • ••

••

-.1

94

8

+.0

47

1

•••••

•••••

-.0

47

1

•••••

•••••

-.0

47

1

+.0

84

8

-.1

94

8

•••••

+.0

84

8

•••••

• ••••

-.0

84

8

+.1

94

8

• ••

••

• ••

••

NO

R M

AL

E

Q

UA

T

IO

N

S •

-.3

08

4

-.O

a0

8

+.2

02

8

+.0

60

8

-.2

52

9 -.

05

69

-.

35

35

-.

03

54

+

.45

57

+

.13

67

+

.09

42

+

. 0

04

7

+.1

00

6

+.0

45

3

+.3

81

5

+.0

63

3

-.5

09

2

-.1

73

1

-.1

47

7 -.

02

38

-.

04

71

-.

02

94

+

.04

77

+

. C

076

-.1

10

0

-.0

35

8

-.0

84

8

-.0

54

3

+.1

94

8

+.1

56

8

The

norma~

equ

atio

ns

may

be

wri

tten

by

insp

ecti

on

as

foll

ows

owin

g to

th

e si

mp

lici

ty o

f th

e co

nd

itio

n e

qu

atio

ns.

To

fo

rm

the

Nor

mal

E

quat

ions

: 1

et.

S

qu

are

each

q

uan

tity

in

'oo

lum

n

1 le

ft

han

d

sid

e

tab

le

of

oo

rrela

tes

an

d

mu

ltip

ly

by

it

s 1

/10

p

,an

d

add

th

e resu

lts:

+0

.72

60

; 2f

t,d.

Mu

ltip

ly

eao

h

qu

an

tity

iq

o

olu

mn

1

by

q

uan

titi

es

in

oo

lum

n

2 o

n

sall

ie

lin

e

mu

ltip

ly

tiy

their

1

/10

p

, an

d

ad

d resu

lts-

-0.2

25

0;

3d

•

Mu

ltip

ly eao

h

qu

an

tity

in

o

olu

mn

1

by

q

uan

titi

es

in

oo

lum

n

3 o

n

sam

e li

ne m

ult

iply

b

y th

eir

1

/10

p

, an

d

ad

d resu

lts:

-0.8

00

0;

4t1

\,.

Mu

ltip

ly

eao

h

qu

an

tity

in

o

olu

mn

1

by

q

uan

titi

es

in

oo

lum

n

4 o

n

sam

e .

lin

e

mu

ltip

ly

by

th

eir

1

/10

p

, an

d

ad

d

resu

lts=

+

0.0

00

0:

5;~lh

Mu

ltip

ly

eao

h

qu

an

tity

iq

o

olu

mn

1

by

q

uan

titi

es

in

oo

lum

n

5 o

n

sam

e "

lin

e

mu

ltip

ly

tiy

their

1

/10

p

, an

d

ad

d resu

ltsa

-0.1

00

0;

~~.~

Mu

ltip

ly

eao

h

qu

an

tity

in

, o

olu

mn

1

by

q

uan

titi

es

in

oo

lum

n

8 o

n

sam

e li

ne"u

ltip

lyb

y th

eir

1

10

p

, an

d

ad

d re

su

lts=

+

0.0

00

0 •

. ~al.~ives

the fir

st

no

rmal

eq

uati

on

. 7

lt,b

;"

Sq

uare

eao

h

qu

an

tity

in

o

olu

an

2

an

d

mu

ltip

ly

by

it

s

1/1

0

p,

and

ad

d

resu

lta=

+1

.02

50

: "tMi

i~'

Mu

ltip

ly

eao

h

qu

an

tity

in

o

olu

.an

2

by

eao

h

ad

jao

en

t q

uan

tity

in

tu

rn

in

the

oth

er

oo

lum

ns

by

t.h:air

1

/10

p

ae

ab

ov

e

an

d

ad

d resu

lts·

r;\:B

,., "

.ro

oeed

as

.bo

v-a

u

nti

l all

·· te

rae

hav

e

been

m

ult

ipli

ed

an

d

t.h

e '6

th

':bera

al

eq

ua\i

on

h

ae

be.

'!!

0Q

.ipi.

,t..e

d.

' '

a b o d e f g h i j k 1 m

n p

NOR ),{ A L E QUA T ION S •

1 :2 ~ 4 5 {: ( q) SUll' Cbeck +0.7250 -0.~250 -0.3000 +0.0000 -0.1000 +0.0000 -0.062 +0.1000 +0.0380 -D-.-2~+1.0250 -0.3000 -0.0500 +0.0000 +0.0000 +0.153+0.4800 +0.6030 -0.-3000 -0.-3000 1+1.1150 -0.1750 -0.3400 +0.0000 -0.434 +o.oeoo -0.4340 +0.-0000 -0.-0560 -o-.1~l+1.5100 -0.1600 -0.3210 -0.157 +0.8040 +0.6470 -0-.-1000 +O-.eOOO -0-.-34-00 -Q.--lOOOl :1! 3850_ -0.1600 +0.204 +0.6250 +0.8290 +0.0000 +O-.-Geoo +o.eooo -0-.--3"21-0 -o.lbOO i~J_.1210 -0.025 +0.6400 +0.612-.Q Note: Sum and C~eck oolumns will not be used in the following aolution of the above normal equations; but will be used later in a subsequent solution of normal equationa.The (q) is the absolute term.

(P) Solution of Normal Equations. 1 :2 4 Ii I) ABSOLUTE

TER II & +0.7250 -0.2250 -0.3000 +0.0000 -0.1000 +O.OOCO -0.0620000 & (-1.3793) +0.3103 +0.4138 +0. 00 CO +0.1379 -0.0000 +0.0855166

l' • • • • +0.9552 -0.3931 -0.0500 -0.0310 +0.0000 +0.1337614 +0.0325 -0.0000 -0.1400348 P. • • · • (-1. 0469) +0.4115 +0.0523

f§ • • • • • • • • • +0.8291 -0.1956 -0.3942 ·+0.0000 -0.4046128 +0.4754 -0.0000 +0.4880035 Ft • • • .' . • • • .(-1.2061) +0.2359

-

'Pr • • • • • • • • • • • • • +1.4613 -0.2546 -0.3210 -0.2454525 (-0.6843) +0.1742 +0.2197 +0.1679631 18 • • • • • • • · . . . . .

-

& • • • • • • • • • • • • • • • • • • +1.1384 -0.2159 -0.0353133 It. • • • • • • • • • • • · • • • • • ·(-0.8784) +0.1896 +0.0310192 -:r11 • • • .. • • • • • • • .. • • • • • • • • • • +1.0096 -O.O856~13 I12 • • • • • • • • • • • • • • • • • • • • • • (-0.9905) +C.0848079

Notes on The P,G,R Solution of Normal Eauations.

Opposite flagella of this thesis will be found the least. SQuare figure adjustment of a Geodetic Guadralateral on the folded plate. The problem cn the plate was taken from the Topographic Instructions of the United States Geological Survey page 61 to page 71,for the instruction of students in my class in Geod~tic Computations. On comparison with the Geological Survey publication in the description of the method for solving normal equations, it will be found that the encluded plate goes much more into the details of the method.

As the ~,~,R, solution of normal equations used here in this level net adjustment is identical with that used in the adjustment of the triangulation scheme,by referring to .tbe plate opposite pagel18, it wilJ be found very easy to follow the process and steps in tb~ give!). computation and also in other computations in tbis thesis where the P,G,B solution of normal equations has been employed.

-52-

(q) Solution of Normal Eouations. 2 1; • 'i 6 ABSOLUTE

ql ++.O~50 -0.3000 -0.0500 +0.0000 +0.0000 +0.15~~OO q2 -O~209§ __ :2~2221 __ ~2!2QQ2 __ :Q!2312 __ ~2~2222 __ :Q~Q~2!2§§

+0.9552 -0.3931 -0.0500 -0.0310 +C.OOOO +0.1337614 -------------~-----------------.......;-'-----qa •••• • +1.1150 -0.1750 -0.3400 +0.0000 -0.4340000

q~ ••••• -0.124'1 +0.0000 -0.0414 +0.00(0 -0.0256556 q, -0.1618 -0.0206 -0.0128 +0.0000 +0.0550428

q. • • q, • • qs • • q,

+0.8291 -0.1956 -0, 3942 +0.0000 -0.4048128 ...-..---------------------------+1.5100 -0.1600 -0.3210 -0.1570000 -0.0000 +0.0000 -0.0000 -0.0000000 -0.0026 -0.0018 +0.0000 +0.0069957

• • • •• • • '-0.0461 -0.0930 +0.0000 -0.0954482 +1.4613 -0.2546 -0.3210 -0.2454525 ------------------------

q iD· • • • • • •. ' . . . +1.3850 -0.1600 +0.204C((0 qu' • -0.0138 +0.0000 -0.0085498 q 11· • . ••• . • -0.0010 +0.0000 +0.0043472

• • • • . -0.1874 +0. COOO '-o .1923529 q 23 •

q if· • • • • .•••• -0.0444 -0.0559 -0.0427578 +1.1384 -0.2159 -0.0353133 ~-------~-----------.---. q 15 •

q •• . . . . . .

q 27· • • • • • • • q.. . . . . . . . . q... . . . . . .. . . . . q.,. • . . . • • . . . . . . . . . .

+1.1210 -0.025((00 -0.0000 +0.0000000 -0.0000 -0.0000000 -0.0000 +0.0000000 -0.0705 -0.0539259 -0 .• 0409 -0.0066954 ---------~----- ----+1.0096 -0.0856213

(r) Solution of Normal Equations. 1 :? 3 4 f5 6

r1 +0.0855166 -0.1400348 +0.4880035 +0.1679631 +0.0310192 +0.0848079 r2 +0.0000000 -0.0000000 +0.0000000 +0.0186323 +0!2~~2Z96 ra +0.0064949 +0.0015307 +0.0223908 ~2!00~~~~ +0.0470988 r4 +0.0000000 +0.0101880 +0.0459533 +0.H~48000 r, +0.2302166 +0.2289370 +0.5563476 r. +0.0312227 :0:1006209 Some authorities on Least --------r, +0.3534508 Square, call Iter) Solution of Normal

Equations" the'l "B~'Ck Solution".

Summary of Junction Elevations in Dawson Township, Phelps County,Missouri~

A = 1087.8000 -----> 1087.8000 - 13.1646 - 25.9044 --------- . -----

B = 1074.6354 + 8.7948 --,.......--..-.-

C = 10B3.4302 + 15.4541 --~

I = 961.8956

D = '1098.8843 ---> 1098.8843 - 24.8794 -177.6017 ------ - .. ---

E = 1074.0049 G = 921.2826 Cheok + 80.0613 --F = 1001.3439 ----->

+ 82.0863 1001.3439 + 40.5103 -----.-

C = 1083.4302 H = 1041.8542 ~-> 1041.8542 Cheok + 32.7811 - 73.4836

B = 10;~'.6353 J = 968.3706 Cheok - 13.7303

954.6403 <,--- M = 95~~6403 +107.2553 - 33.3578

A D JUS TED

A = 1087.8000 B = 1074.6354 C = 1083.4302 D = 1098.8843 E = 1074.0050

I = 1061.8956 Cheok

E LEV A T ION S •

F = 1001.3439 G = 921.2829 8 = 1041.8542 I = 1061.8956 J = 968.3706

M = 954.6403.

G = 921.2825 Cheok

Points K, L, N, 0, P, are adjusted as intermediate points and take corrections proportionate to their distances from adjacent junction points.

ADJUSTED ELEVATIONS. K = N = L = ° =

p = Points Q, R, S, are at the ends of spur lines and will take the same corrections as P, N, O,respectively.

ADJUSTED ELEVATIONS. Q= R= S=

A b

rid

g e

d·

Met

:h 0

d

A.

N.

0 R

11

A

LE

Q

UA

T

IO

N

S

• C

on

t' d

•

A

. B

CD

.

E

F

Co

nst

. ci

he~k

.~q

No

te.

(I)

+0.

7250

-o.

2~50

-0

.30

00

+

0.06

00 -

0.1

80

0

+0.

0500

-o.

0~20

+0.

0~08

It

m~st b

e re

mem

bere

d th

at th

e(Ch

~Ok)

(2

) •

• •

• +

1.02

50

-0.3

00

0 -

0.0

50

0

+0

.00

00

+

0.00

00

+0.

1530

+

0.60

30

colu

mn

lnclu

des

all

o

f th

e co

eff

icie

rits

· (

3)

• •

• ••

• +

1.11

50

-0.1

75

0 -

0.3

40

0

+0.

0000

-0

.43

40

-0

.43

40

o

f an

eq

uat

ion

, wh

eth

er w

ritt

en

o

r n

ot,

so

(4)

• •

• •

• •

• •

••

+1.

5100

-0

.16

00

-0

.32

10

-0

.15

70

+

0.64

70

that,

whe

n th

e ab

rid

ged

fo

rm

is

use

d,a

s (5

) •

• ••••••••

• •••

+1

.38

50

-0

.16

00

+

0.20

40

+0

.82

90

ju

st

to

the

left

, th

e co

eff

icie

nts

m

ust

(8)

.•••.••

• •

• •

• +

1.12

10 -

0.0

25

0

+0.

6150

be

re

ad

dow

n an

d to

th

e ri

gh

tjio

e:

L 1

ine(2

);

add

colu

mn

2 w

ith

li

ne(

2)=

+ ~030

cr p

\\ $0

/&/ f-

Ion-

-~

So

luti

on

o

f N

orm

al

Eq

uat

ion

s by

A

bri

dg

ed

Met

hod.

.0

•

A,/.

( B

''\

C

D

E

F

Co

ne

t.

Che

ck.

Pro

oe

ss

. (2

) •

'.'

• +

1.02

50 -

0.3

00

0 -

0.0

50

0

+0

.00

00

+

0.00

00

+0.

1530

+

0.60

30

(Nor

mal

E

qu

atio

n

(2).

N

ote:

: T

hruo

ut

(7) •••• ::

2.:.

Q~~~

~~~2

.:.2

222_

:2!.

2~~2

_~2.

:.2~2

2-:2

:.2~~§=

!2.:.~!~§

(I)x

EJt

f)

or

x 0

.31

03

. ~~

:tfP

~~~e

88

(II)

' ',

' .

+0

.95

52

-0

.39

31

-0

.05

00

-0

.03

10

+

0.00

00

+0.

1338

+

0.61

48 ~

(I

hor.

O.J

14

9)

(2)+

(7).

n

um

erat

or

of

---.---......----~ ...

... -~----------------------

-ea

oh

fr

ao

tio

n

(3)'

"

••••

• ·.

+1

.11

50

-o.~750 -o~3400

+0.

0000

-0

.43

40

-0

.43

40

(N

orm

al

Eq

uat

ion

(3

).

use

d.a

e.a

(8

) •••••

• ••

-0.1

24

1

+0.

0000

-0

.04

14

+

0.00

00 -

0.0

25

7

+0.

0157

(I

)x(+

:fii

) o·

r x

O.4

13

8.

i~!t!pl1er

has

· (

9) •••••••.

-0.1

61

8 -

0.0

20

6-0

.01

28

+0

.00

00

+

0.05

51

+0.

2530

(I

I)x

(+=

,fH

i)

or

x 0

.41

15

ig

n

oh

ang

ed.

'(II

I)

'.'

•.••

+0:8

29i-

:O:1

956-

:O:3

942-

~O:OOOO -

:0:4

046-

:0:1

653

1(2

h~r .

C.1

05

3)

(3)+

(8

) + (9

) •

(4

)'

••

• •• ~-:-:--:'i:5ioo-:o:i60C;--0:32io:O:'i570~O:647C

(Nor

mal

E

qu

atio

n

(4).

(1

0)'

•

• ••

• ••••

+0.

0000

+

0.00

00

+0.

0000

+

0.00

00

+0.

0000

(I

) x

(+',~",)

or

x 0

.00

00

. (1

1)'

•

• •••.•••

-0.0

02

6 -

0.0

01

6

+0.

0000

+

0.00

70

+0

.03

22

(I

I)x

(+:m

,)

or

x 0

.05

23

. (1

2)"

••••••••

-0.0

46

1

-0.0

93

0

+0.

0000

-0

.09

54

-0

.03

S0

(I

II)x

(+:f

lt\)

o

rx 0

.23

59

. -----------------------

--------

---

t--

. (I

V)"

•

• •

• •••.

+1.

4613

-0

.25

46

-0

.32

10

-0

.24

54

+

0.64

02 v

'(L

hor.C.040~·n

(4)+

(10)

+(1

1) +

(12

).

'(5

)"

••

• •

-:-:-:-~

1:'3

850-

-0:1

600:

O:20

40 :

0:8

29

0

(Nor

mal

E

qu

atio

n

(5).

(1

3)'

•

. •••••.•

" •••

-0.0

13

8

+0.

0000

-0

.00

85

+

0.00

52

(I)x

(+ff

i%)

or

x 0

.13

79

. (1

4)"

•••

'.'

•••

• ••••

• •

-0.0

01

0

+0.

0000

+

0.00

43

+0.

0200

(n

)x(+

:HH

) o

rx 0

.03

25

. (1

5)'

" ••••

• •

• •

0 •

0 •

-0.1

87

4

+0.

0000

-0

.19

24

-0

.07

86

(I

1I)

x(+

:Htf

)orx

0.4

75

5.

· (1

6)'

•••

• •••

" ••• -o~2~~~_:

_9.:

2~2~_:

2~2~

~7_~~~

111E

(I

V)x

(+'l

li-i

.,)o

rx 0.174~.

(V)·

••••••••.•••• ~± .13~i-=0 .2~~~:2.:.2~~~_~2.:.~~Z~

v'(L

h

or.

O .8

87

2)

(5)

+ (1

3)+

(14

)+ (1

5) +

(6).

(6

) •••••.

'.'

0.

' ••••••

• •••

+1.

1210

-0

.02

50

+

0.61

50

. (N

orm

ttl

Eq

uat

ion

(6

).

(17

)'

J'"

•••

" •

• +

0.00

00 -

0.0

00

0

+0.

0000

(I

)x(o

f,"i

3)

or

x 0

.00

00

. (1

8)'

•

••

. •

• •

• •

• ••

+0.

0000

-0

.00

00

+

0.0

(00

(I

I)x

(+:"

")o

r x

0.0

00

0.

(19

)'

• .•..

• •

• •

• .

. .

. •

+0.

0000

-0

.00

00

+

0.00

00

(III)x(of,fli~)orx

0.0

00

0.

(20

)·

• •.

•.•••

• •

• •

• •

• •

• -0

.07

05

-0

.05

39

+

0.14

07

(IV

)x(+

'tH

-ii)

or x

0.2

19

7.

(21

)" ',

' '.

' •••

','

• •

• •

• ••

• -0

.04

10

-0

.00

67

+

0.16

83

(V)x

(+-t

tiH

)or

x 0

.18

97

. (v

I)"

'. •

•• •.

• ••

..

',.

• . S~~~:2~§~. !0.9~ I(

r ho

r.O

. 82

39

) (6

) + (1

7)+

(18

) +

(19

)+(2

0)+

(2

1).

rl~ 1

\ C

o

R

R

E

L

A B

RID

G E

D

AT

E

S •

ME

T H

OD

A

•

AB

C

D

E

F P

R P

C F

S

S •

Co

nst

ants

+

0.0

62

0 -

0.1

33

8

+0

.40

46

+

0.2

45

4

+0

.03

53

+

0.0

85

6 •••

co

py

(V

I)(V

)(IV

)(II

I)(I

I)(I

)C

on

st

Col

umn

----

-=

. +

1.0

09

5··

·Div

ide(V

I)C

on

stC

olum

a b

f(V

I)F

Col

umn=

+O.0

848

F

-ter

ms

+0

.00

00

+

0.0

00

0

+Qj .

00

00

+

0.0

27

2

+0

.01

83

•••

••••

••••

I)d

ult

i p

ly (V

) (I

V)(

II 1

)( 1

1)(

I)

F C

ol.

)(+

0. 0

84

8

---

---

+0.C

8!.1

.B=F

P.

1ao

e· a

t h

ead

o

f C

ol.

e

Tab

le

of

Co

rrela

tes.

+

0.0

53

6 •••

•••

; ••

•• ~

oolu

mn

E.

E-t

erm

s

D-t

erm

s

- C-t

erm

s

B-t

erm

s

~:i384~ •

••••

••••

• O

ivid

e ~

Co

l.E

bf(

V)C

ol.

E=+

C.0

4?0

8

+0

00

47

+0

0

01

5

+C

01

57

+0

0

12

0 ••••••••••••••••••• M

uit

iply

(IV

)(II

I)(I

I)(I

)Co

l.E

x

+O

.04

?1.

'.

. •

_.:

. __

__

+f

'" f"'

4rC

fJ=

"' ••

••••

•• ~laoe

at

hea

d

of

Co

l.5

T

able

o

f C

olt

rela

tes.

\.

.•

\..

, v

..,

1 D

+

0.2

84

6 •••

•• ~.............

00

um

n.

. --

----

-:

+1

.46

13

••••••••••••••••••• 0

ivid

e ~

Co

l.O

by

(IV

)Co

l.D

=+0

.19

46

. +

0.0

00

0

+0

.00

97

+0:2~~~ •

• J.! •

• Q

! ••

••••

••••

••••

••••

Mu

ltiP

ly(I.

II)

(II)

(IlC

ol.

D )

( +

0.1

94

8.

+\.i

.l94\

..i-D

••••••••••••••••• ~laoe

at

hea

d

of

Co

1.4

T

able

o

f C

orr

ela

tes.

+

0.4

58

4 ••••••••••••••••.

.•••

••••

••••

L

oo

lum

n C

. ~-....-.::

+0

.82

91

••••••••••••••••••••••••••• 0

ivid

e ~

Co

l.C

by

(III

)C6

1.C

=

+0

.55

29

. +

0.1

85

9

+0

.21

73

)d

ult

iply

(II)

(I)C

olu

mn

C

x

+0

.55

29

. -

---

+0

5529

=C

••••••••••••••••••••••• ··~

laoe

at

hea

d

of

Co

l.3

T

able

o

f C

orr

ela

tes.

+O.094~ •

•• :

•••••••••••••••••••••••••••••••

oolu

mn

B.

-=

+

O.9

552

••••

••••

••••

••••

••••

••••

••••

••••

•••

Div

ide

~ +0~0223 •

•••••••••••••••••••• ···

····

····

····

····

···M

ult

iply

--

----

-+C

.099

1=S

•••

••••

•••.

•.••

••••

.•••

••••

••••

•• E

lao

e

at

+0

.25

49

....

....

....

....

....

....

....

....

....

Io

olu

mn

---

---:

-.

... .

. +O~7250

••••••••••••••••••••••••••••••••••••••••••• Di~

ide

I

Co

l.B

by

(II)

Co

l.3

= +

0.0

99

1.

(I}

Co

1.B

x

+0

.09

91

=+

0.0

22

3.

hea

d

of

Co

l.2

T

able

o

f C

orr

ela

tes.

A

. C

ol.

A b

y (I

)Co

l.A

=

+0

.35

16

.

+O.3

518=

A ••

••••

••••

••••

••••

••••

••••

••••

••••

••••

••• P

lao

e

at

hea

d

of

Co

l.1

T

able

o

f C

orr

ela

tes.

(R

efe

ren

ce

1s

mad

e ab

ov

e b

etw

een

th

is p~ge

and

the

pag

e p

reo

eed

-in

g

th1

S.)

.

The

m

etho

d g

iven

ab

ove

is

a su

bst

itu

te f

or

the

(P),

(q),

(r)

So

luti

on

o

f N

orm

al

Eq

uat

ion

sj

all

o

ther

st

eps

in

the

pro

cess

o

f le

vel

n

et a

djust~

men

t ar

e id

en

tical

wit

h

tho

se

giv

en

in P

art

of

this

T

hes

is,

and

wil

l n

ot

be

giv

en a

gai

n

her

e.

Not

e:

the

I (C

heck

m

ark)

at

the

ends

o

f ro

ws

(II)

,(II

I),(

IV),

(V),

an

d(V

I),

are

wri

tten

to

in

dic

ate

th

at

the

I o

f C

olum

n an

d Ro

w

are

exac

tly

sati

sfie

d.

Una

void

able

dis

crep

anci

es

in

the

last

fig

ure

o

f th

e Check~term,due

to re

m

ain

der

s,sh

ou

ld b

e re

mov

ed

by

arb

itra

irly

cor~ctiDg

the

check~term

bef

ore

p

rece

sain

g w

ith

th

e n

ext

step

in

th

e eli

min

ati

on

;th

is i

s b

est

done

by

d

raw

in

g a

li

ne

thru

th

e er

ron

eou

s fi

gu

re

and

wri

tin

g

the

co

rrect

one

abov

e it

.' A

B R

ID G

E D

M

ET

HO

D

B

• N

OB

)l

A L

E

QU

A

T I

ON

S

• /

1 2

34

n

6 N

l,

1 +

0.7

25

0

-0.2

25

0

-0.3

00

0

+0

.00

00

-0

.10

00

+

0.0

00

0

-0.0

62

0

+0

.03

80

N

ote

. 2

•••

• +

1.0

25

0

-0.3

00

0

-C

. 0

50

0

+0

.00

00

+

0.0

00

0

+0

.15

30

+

0.6

03

0

It

mu

st

be

rem

e'm

ber

ed

that

the (

Ch

eok

E)

3 •

• •

• •

• •

• +

1.1

15

0

-0.1

75

0

-0.3

40

0

+0

.00

00

-0

.43

40

-0

.43

40

o

olu

mn

in

olu

des all

o

f th

e

0 ttl

i t

4,

• •

• •

• •

• •

• •

• •

+1

.51

00

-0

.16

00

-0

.32

10

-0

.15

70

+

0.6

47

0

an

eq

uati

on

,wh

eth

er

wri

tten

g:

no~

:: :h

:f 5

• •

• •

• •

• ••

• •••

• •

• +

1.3

85

0

-0.1

60

0

+0

.20

40

+

0.8

29

0

wh

en

the

ab

rid

ged

fo

rm

is

use

d

th

' f

' 6

• •

• •

• •

• •

• •

• •

• •

• ••

• •

• +

1.1

21

0

-0.0

25

0

+0

.61

50

fioi

~ntB

m

ust

b

e re

ad

, do

wn

and'

toet

~~er

ight

. S

olu

tio

n

of

No

ra.l

E

qu

ati

on

s b

y

Ab

rid

g;a

eMe

't&ga

~ o

f ch

eck

o

po

sit

e

lin

e

(14

).

•

1 2

3 4

5 6

N

2. Pr9c~ss

. '

1)

+0

.72

50

-0

. 2

25

0 -O

~ 3

00

0

+0

.00

00

-0

.10

00

+

0.0

00

0

-0.

C62

0 +

0.0

38

0

No

rmal

E

qu

ati

on

\ 1

} .•

I)

•

• •

• +

0.3

10

3

+0

.41

38

-0

.00

00

+

0.1

37

9

-0.0

00

0

+0

.08

55

-0

.05

24

D

ivid

e

No

rmal

E

qu

ati

on

(1

) b

y

-0

.72

50

.

7 •••• -0

.06

98

-0

.09

3-1

+

0.0

00

0

-0.0

31

0

+0

.00

00

-0

.01

92

+

0.0

11

8

uIt

.No

rmal

• (1

) x

+0

.31

03

. 2

1

• •

• •

+1

.02

50

-0

.30

00

-0

.05

00

+

0.0

00

0

+0

.00

00

+

0.1

53

0

+0

.60

30

N

orm

al

Eq

uati

on

(2

).

8 ••••

+0

.95

52

-0

.39

31

-0

.05

00

-0

.03

10

+

0.0

00

0 +

0.1

33

8

+0

.61

48

/ !

(2)

+(8

). ff

hori

zont

~lly

'+0

.61

49

).

I ) ••••••••

+0

.41

15

4+

0.0

52

35

+0

.03

24

5+

0.0

00

0

-0.1

40

08

-0.6

43

63

D

ivid

e

(8)

by

-0

.95

52

. 3

) •

• •

• •

• ••

+1

.11

50

-0

.17

50

-0

.34

00

+

0.0

00

0

-0.4

34

0

-0.4

34

0

No

rmal

E

qu

ati

on

(a

l.

9)

• •

• •

• •

• •

-0.1

24

1

+0

.00

00

-

0.0

41

4

+0

.00

00

-0

.02

57

+

0.0

15

7

Mu

lt.N

orm

al

Eq

.Ji)

x

+0

.41

88

. 1

0

• •

• •

• •

• •

-0.1

61

8

-0.0

20

6

-0.0

12

8+

0.0

00

0

+0

.05

51

+

0.2

53

0

ult

(8

) x

+0

4

15

4.

11~·

•

• •

• •

• •

+0

.82

91

-0

.19

56

-0

.39

42

+

0.0

00

0

-0.4

04

6 -0.1e53~

~ (

3'+(

9)+(

10~.

h

ori

zo

nta

lly

-0

.16

53

).

rli)

• •

• •

• •

• •

• •

• +

0.2

35

91

+0

.47

54

6+

0.0

00

0

+0

.48

80

0+

0.1

99

37

D

ivid

e

(11

) b

y

-0.8

29

1.

4) ••••••••••••

+1

.51

00

-0

.16

00

-0

.32

10

-0

.15

70

+

0.6

47

0

No

rmal

E

qu

ati

on

(4

).

12

1 ••••••••••••

-0.0

00

0

+0

.0(0

0

-0.0

00

0

+0

.00

00

-0

.00

00

M

ult

.No

r)m

al

Eg

.(!)

x

+0

.00

00

. I

13

. ••

..

, ••••

-0.0

02

6

-0.0

01

6

+0

.00

00

+

0.0

07

0

+0

.03

22

M

ult

. (8

x

+0

.05

23

5.

\.J1

14

•

Wor

m o~

C

heC

K.

. ••

...0

.04

A1

-0

.09

30

+

0.0

00

0

-0.0

95

4

... 0

.03

90

, u

lt.

(11

') x

+0

2

59

•

~ 1

5 ••

+1

.46

13

-;-~~~~ +

1~461$ -

0.2

54

8

-0.3

21

0

-0.2

45

4

+0

.64

02

v ~

(4)+(12)+)(13)+f14t.(~ h

or.

+0

.64

03

).

IV ••

-0

.25

46

•••••••••

+0

.17

42

3+

0.2

19

67

+0

.16

79

3-0

.43

81

0

Div

ide

(15

b

y

... 1

.46

13

. )

-0

. 3

21

0

1 tl

>

t i

& J

5

••

-0

.24

54

•••••••••

+1

.38

50

-0

.16

00

+

0.2

04

0

+0

.82

90

N

orm

a ~qua

on

5

.

18

l ••

+0

.6

4m

3 ~HorizontBl1y

· ••

-0.0

13

8

+0

.00

00

-0

.00

85

+

0.0

,05

2

MU

lt.f

No

)rm

al

Eg

. 1

x +

0.1

37

9.

17

••

+0

.6

40

2 ~Column •••

. ~ •••

-0.0

01

0

+0

.00

00

+

0.0

04

3

+0

.02

00

M

ult

. 8

x +

0.0

3

45

. 1

8 ••

0 0

00

1 err

or

••••••

-0.1

87

4

+0

.00

00

-0

.19

24

-0

.07

86

M

ult

. 1

1)

x +

0.4

75

46

. 1

9

• •

• •

••

• •

: ••••••

-0.0

44

4

-0.0

55

9

-0.0

42

8

+0

.11

15

/ M

ult

. 1

5)

x +

0,1

74

2a.

20

•

• •

• •

• •

• ••••••

+1

.13

84

-0

.21

59

-0

.03

54

+C

fr.:c

S7

1I

L. (~)+(1(6)+)

(17

}+

(18

)+(1

9).

(~

ho

r.+

0.8

87

1).

V

) •

• •

• •

• •

• •

• •

• •

• •

• +

0.1

89

65

+0

.03

11

0-0

.77

92

5

D1

v1

de

20

b

y

-1.1

38

4.

8) ••••••••••••

• •

• •

• •

• •

+1

.12

10

-0

.02

50

+

0.6

15

0

No

rmal

E

qu

ati

on

(6J.

21

. •

• •

• •

• •

• •

• •

• •

• +

0.0

00

0

+0

.00

00

+

0.0

00

0

Mu

ii.

N(o

rmal

E

q.(

l x

+0

.00

00

. 2

2

• •

• •

• •

• •

• •

• •

• •

• •

• •

+ C

. 0

00

0

+0

.00

00

+

0.0

00

0

Mu

.'

81

x

+0

.00

0

•

-26

•

• •

• •

• •

• •

• •

• •

• •

• +1

,009

6 -0

.085

6 +0.9

2a8~ ~ t

All

,s )+(

lil.'~I'

+(24)+(2

5)(z,+o.

92.0).

23

•

• •

• •

• •

• •

• •

• +

c. 0

00

0,

+0

. 0

00

0

+0

.00

00

M

u 1 t.

1

1

x +

0.0

00

0.

' 2

4 •••••••••••••••••

• •

• -0

.07

05

-0

.05

39

+

0.1

40

6

Mu

lt.

15

x

+0

.21

96

7.

25

•••••••••••••

• •

• •••

• -0

.04

09

-0

.00

67

+

0.1

68

2

u x

l.

VI •••••

• •

• •

• •

••

• •

• •

• ••••••

+0

.08

47

9-0

.91

50

2

Div

ide

(28

) b

y

"'1

.00

96

. B

A

C

K

SO

L U

TI

0 N

•

12

3

4 n

6 +

0.0

85

50

-0.1

40

08

+0

.48

80

0+

0.1

67

93

+0

.03

11

0+

0.0

84

79

••••••••

+0

0

00

00

-0.0

00

00

+0

.00

00

0+

0.0

18

63

+0

.01

60

8 ••••••••

• ••

• +0:00849+0.00153+0.02239+0~00800+0.04718.

• ••••••

• +

O.O

OO

OO

+0

.01

01

9+

0.0

.5g

e+O

.19

47

8 •••••

• ••••••••

• •

+0

.23

02

2+

0.2

28

94

+0

.55

83

4 •••••

~ •

••

• •••••

+0

.03

12

2+

0.1

00

58

r ••••••••

• ••••••

• •

• •

••

+0

.35

34

3 •••••••••••

• •

• •

• •

• •

••

• •

• ••

• •

Co

py

V

l)4

(V

), \

IV)(

(IIll' (If)

' (I)

Co

l.N

. M

ult

. V

),d

v)t.

IIIJ6(

II ,(I

co

)1(6

)X+

0.0

84

79

. M

ult

. IV

) (Ill

'il

l, I

Co

l 5

x+

0.0

47

18

. M

ult

. IIl~l.(lI

, I)

0

um

n(4

x

+0

.19

47

6.

Mu

lt.

II)L

{I)

CO

u

,n

(3)

x +

0.5

56

34

. M

ult

. I)

u

olu

an

(2

) x

+0

.10

05

8.

~ C

olu

mn

(1

).

+ + +.

ADJUSTMENT BY THE METHOD OF CORRELATES. THE 0 R Y •

The final adjusted values of the observed quantities must satisfy the conditions of the problem in hand, and must be the most probable values,acoording to the Theory of Least Squares, that will satisfy all the conditions. Therefore,the sum of the weighted squares of the corrections(residuals,v) must be a minimum.

Then: U =W:J.ff +v&. b2 + 1't3 c2 + Nc. d!- ••••• . + V+1 rz! = a minimum

which ~ust be satisfied simultaneously with the following conditions. Multiplying the condition equations on page 50 in suc.cession by the factors, -2~, -2~, -203, etc., respectively,

+2Ot Ch. + 2Ct d + 20t c +2Ct b +2Ct a = 0 -2G< ~ -2~ g + 2~ f+2(1 e -2(1 c = 0 +2Ce ce -203 e 2<1 h +2Cb i -203 b = 0 .. . .. . ... .. . ... .. . .. . ... ... . -20; qs -20; i -2Cs j +20; 1 +2Cs k -20; a = 0

(2)

Adding these equations to (1) and collecting the coefficients of the separate a, b, c~ etc., we have the requirement that,

?il.tf + 2a(Ct - Os ) +

Vi2 b2 + 2b (Ot - 03 ) + (3)

VI5 c2 2c(~ + Ct ) +

. . . . . . . . . + + 2(Ct.Q1-CeQ2+Ce~+ ••• ) = a minimum.

For the minimum, the first derivative of this expression with respect to each a, b, c, etc., must b~ placed equal to zero. Therefore,

dU = 2W;t a + 2(Ct - Cs) = 0, dividing by .2 and solving for a, da 1

dU - 2W:l b + 2 (~ - Cb )

db

a = - (-Ct. + Cs), W1

= 0, dividing by 2 and ~olving for b,

b =. ! (-OJ. + Ce ), Yi2

-58-

dU 2~ Y -2((1 c,.) dividing = + = 0, by 2 and solving for c, dc

1 (-0,. ~) c = - +

~

dU = . . . . . . . . = 0, etc. dz

Where W1, Vi2, ~, etc.= ~ are the weights respectively, d= distances U = ~ weighted residuals squared, desired to be a minimum, a, b, c, etc. = corrections for the links, c,., Cl2, Cs, etc.= the adjustment factors solved for in the

normal equations. ~, CJ2, ~, etc.=constants of observed closure respectively

in the several circuits.

~~ = the differential of U with respect to a, or the first derivative.

The constant 2 comes from the differentiation of the expressions in each equation equated to O.

substituting these values of the a, b, c, etc. in the condition equations on page 50, and combining the coefficients of ~, Ck, Cs, etc.,we obtain the Normal Equahons: Which partake of the following form:

aa , [w]~

[a:]~ [~] OJ. w

+

+

+

[a:]~ + r~]Ca bb

(b:] Ca Iw)~ +

[bc]~ w + [~]Cs w

+ . . . . + qi = 0

+ + ~ = 0 (4)

+ . . . . + Cb = 0

If the foregoing substitutions and and like Cs combined.,we arrive at the normal equations as appear on the top of page 52.

In forming the normal equations, it may be found somewhat simpler to follow the rules of thumb formulated from the foregoing theory. which will be found on page 5~ for the formation of normal equations from the condition equations, and on pages 10 apd 41,by a similar demonstration, for the formation of normal equations from observation equati~ns.

-59-

APPLICATION OF THE THEORY. Least Squares computation consists in rendering the val~e

of U in the following expression a minimum~

o = a2 + .333b2 + • 444c2 + d2 + .333e2 + 2£2 + .222f + • 571h2

+ .294i2 + .. 625j2 + .16Ir + .62512 + • 312nf + .156rr + .124J!.

Condition Terms. -2Ct. [-.062- d - c - b - a] -2~ [+ .153 + g - f - e + c] -2<1 [-. 434 + e + h - i + b]

-2G.. [-. 157 - h + f + p - ill - j]

-2Cs [+.204 + a + i + j - 1 - kJ

-205 [-.025 + m - n + 1]

The coefficients of the above co~dition terms is taken arbitrarily as -2 in order to simplify the expressions for the "V" corrections (ie. a, b, c, etc.) for the links.

To make the above expression for U a minimum requires that its differentials with respect to a, b, c, etc. be each equated to zero, as follows:

2 a + 2Ct. - 2Ce = 0, then

I

a = - Ct. + Os ........ = 1 times (-Ct. + as )

.666b + 2Ct. - 2Ce = 0, then b = - 3Ct. + 3<1 ••••• = 3 "( -Ct. + <1 )

.S8Sc + 2Ct. 2~ = 0, then c = -2.25Ct. + 2.25~ .= 2.25 II (-Ct. + ~ )

2d + 2Ct. = 0, then d=-~ .. , ............ =l

.666e + 2<il - 2Ce :: 0, then e- = - 3~ + 3<1 ........ = 3

"

" ( -Ci + Q, )

-60-

These expressions when simplified and with the Cs transposed to the right-band side of the equations, will then appear as extended to the left, and form the correlates as they are tabulated on page 51.

4f + 2~ - 2(4 = 0, then f = - .5~ + .5~ .......... -.

.444g - 2~ = 0, then g = + 4.5~ .......................

L142h - 2~ + 2(4 = 0, then h = +L75~ - 1..75(4 •••

.. 588i + 2~ - 20s = 0, then i = - 3 .. 4(1 + 3 .4~ ••••

1.250j + 2(4 - 20s = 0, then j = - 1.6(4 + l.6Qs .......

.. 32k + 20s = 0, then k = - .. 625~ " ....... " .......... -

1.2501 + 2~ - 2~ = 0, then 1 = - 1.6Qs + 1.;6~ .......

.. 624m + 2(4 - 2~ = 0, then m = - .. 321(4 + .321~ ...

.. 312n + 2Qs = 0, then

.248p - 2(4 = 0, then

= 0.5

= 4.5

= 1..75

= 3.4

= 1.6

= 6.25

= 1.6

= 0.321

p = + .805(4 .............. = 8.05

-61-

times (~ + ~)

" (~ )

n ( +(1 - ~)

II (~ + Cs)

" (-(4 if- Cs)

"

11 (~ + ~)

fI (~ + ~) \

fI +~

n +(4

PAR TV. - -PROPORTIONAL AD·JUSTMENT OF INTERMEDIATE BENCHMARKS

FROM CORRECTED ELEVATIONS OF ADJACENT ' .JUNCTION PoINTS.


OF

INTERMEDIATE BENCHMARKS

IN THE

DAWSON-ST. JAMES TOWNSHIPS LEVEL NET~

PHELPS eOUNTY~

MISSOURI.

(BA'E~ ON THE LINE O~ LEVELS ~UN 8Y THE U. S. GEOLOaJeAL SU~VEY IN 1908 ALOMG THE FRISCO R. R.

11TH ~E •• AN£NT BENCHMARKS -A- AT ST. dAMES, AN. -E- AT KNOBVIEI).

-62~

I "" \.>I I

!dju

stm

ent

of i

nter

med

iate

ben

ch m

arks

be

twee

n p

oin

t! a

t st

. jam

es

and

po

int

I at

Mil

es S

ehoo

l on

R

epub

lic

Roa

d.

Th

. ad

ju.t

•• n

t o

f th

e •

•

In

ter.e

dl.

te

ben

oh

•• r

k,

oan

b

e .f

f.c

ted

br

eo

all

",

the .dj~.t •

• f

tt

oo

rr.o

tlo

ns

fro

. th

e

dl.

gra.

b.l

ow

an

d

eu

btr

act

In,

.ao

h

oo

rr.o

tlo

n

fro

a

the

ob •

• r

r.d

.I

,vatl

on

a.

It

ap

p.a

r.

In

the

' l.

v.1

n

ot •• ,o

r"If

deelr

ed

, t~e

ad

juet.

,nt

oo

rreo

tlo

ns

oan

b

e oo.p~ted

br

th.f

oll

o.f

ng

f.r

au

l.

for

thl ••• 0

tlo

n,(

A

to

I).

' D

ER

IYA

TIO

N

OF

AD

JUST

ME

NT

FO

RM

UL

A.

E I. v

•

A •

10

87

.89

0

E I. v

I

• •

l061.92~O

Ob

•• r

. O

or

• •

0.0

00

•

I • ·_!~!!:!21!_AdJu •

• \I

)

/'

Ie

~ -

()'O

29

6

Co

r.

R,

xll

IJ

j i.--.-

<.(~

, ~

6,Z5Mi

,~

Cor

rect

ion

c.

for

any

inte

rmed

iate

poi

nt a

t d

ista

nce

d.

from

A

and

be

twee

n p

oin

ts A

and

I:

L

0.02

96 x

d

<:. .

0

C =

... --

----

----

or =

-o.

oo47

36d.

\\

w

6.

25

\ .

E

\..

R

~ I,)

•

1('

\

'1 0

,... ~

"-~

I"-

IC'\

C\I

\()

1('

\ N

0

<!>

.,. N

N

0

• Q

K

'\

... N

0

. ...

0 C<

I 0

• 0

-0

-N

0

. ~

I"-

... 0

• IC

'\

" 0

• C

C\

0 .

(\I

.... •

00

K'I

0

--co

r-

I •

.,. ....

co

· 0

0

0 .

... 0

. "

"'-/'~

-'

A

'0

p

• · .. ... (I)

.. A

0

J U

<I

f ...:

I ~

r-I

c.'J

., ••

• II:

> \£

'\ IC

'\ .0

IC

'\ I"

-(\

I l"

-

-.. •

• •

• ••

0

Pi

(\I

N

QR

APH

F

OR

TH

E A

DJU

ST

ME

NT

D

F

S T

ED

1'

\'\

IC'\

(\I

LIN

e

• ~

lC\

'-0

IC'\ ,...

o o IC

'\ N

)C

.c-- .,. IC'I

00

aa ,...

• ••••

1('\

1('\"''''''''''

INT

ER

ME

DIA

TE

P

OIN

T

BE

TW

EE

N

A

0'\

0

r-I

r-I

r-I

co

IC'\

IC'\

co

N

I"-

;, •

• IC

'\ IC

'\ IC

'\

AN

D

I •

\()

0\

N <:> • H IC'\

(\I •

\()

Merimec -Sprines ~uadrangle. (Latitude 37 45'-38;10ngitude 91 30'-91 45 1 )

(St. James Township)

Line from st. James,northwest via Jefferson road to the intersection of Republic road, -thence north via Republic roqd, 9. 5 miles to the Phel p s-Gasconade Coun·ty Line.

A. St. Jalnes, corner of Meramec and Wasl1ington Sts.,about 70 ft.R. of,center line of Frisco R.R. track ,25 ft. from corner of Commercial Hotel, in angle between cro ss walk and side Walk; iron post stamped "1098 n. ----------- 1087.800 0.000 1087.800

\ i - .

L. st. James, 0.5 miles N.W. of, on Jefferson road at intersection of Parker Lane road, concrete . cu1vert,north end wall, top of; chi se1ed square. -------- ------ 1078.427 -0.002 1078.425

K. st • J arne s , 1. 75 mil e s N. W .0 f , on Jefferson road at intersection of Republic road,on south side of road, 10 n bl ack jack tree, in base of;copper nail and washer. 1092.694 -0.008 1092.686

1 St.Ja.mes,2.25 miles N.of,on Republic road, Jefferson road,O.5 miles N.of,culvert,west end of, sandstone slab, top of;painted square. ----------------------- 1034.345 -0.010 1034.335

2 Jefferson road,l.O mile N.of,on Republic road,T road east at Coxts house,lOO ft. E. of, sand stone boUlder in northeast corner of T road;top · corner broken off and painted white. -------- 1024.715 -0.013 1024.702

3 Jefferson road,1.5 miles north of, R6binsons Creek,O.75 miles south o£,Fitzenriederts residence,100 feet west of,wagon gate post,in base of;copper nail and washer. 998.469 -0.016

-64-

998.453

4 Robinson's Creek,O.25 miles south of,on Republic road,on east side of road,lO If post oak tree,near Kenndyts mail box, in base of tree; nail. ____ ~ ___ ~_---------------w-- 923.575 -0.018

5 Jefferson road, 2.25 miles north of, on Republic road,Robinsons Creek, west bank of,24" white oak tree, in root of;C01)per nail end washer. 917.604 -0.019

(Dawson Township)

6 Robinsons Creek,O.25 miles north 9f, 15 feet west of road,sandstone boulder on 'St. James-Dew son Township line, opposite wire fence corner on east side of road, point is on south hill side 300 feet south of wood chopper's cabin;painted cross on top of

923.557

91? .585

boUlder. ------------------------·· ··-1022.394 -0.020 1022.374

7 Robinsons Creek,075 miles north of, on Republic road,township line,0.5 miles north of,at north end of ridge, 30 feet west o~ raad,se~dstone boulder, top of;chiseled square. ---------- 1039.423 -0.021 1039.402

8 Robinsons Creek,l.O mile north of, on Republic road,at foot of hill, concrete cUlvert · across road,west end wall, top of;chiseled square.- 934.325 -0.023

9 Robinsons Creek,1.25 miles north of, Miles School House,l.O mile south of, T road west,center of roads;road

934.302

elevation. ------------------------ 976.005 -0.024 975.981

10 Miles School House, 0.5 miles south of, on Republic roeA,Marrs Cemetary,400 feet north of,east side of road, sand stone boulder at rail fence corner, top of; chiseled square. -----------1059.0117 -0.025 1059.052

It St.James,6.0 ·miles N.of,on Republic Rd.,Safe,2.5 miles E.of,cross Rds.at Miles School ,N.E. corner of School ~ouse,limestone wheel' guard, top of; pa.inted square. ---... -------------- 1061.925 -0.030 1061.895

-65-

Kdjustment of peint P en Republie Road 0.5 miles south of Ha.rris.a Seh(!)ol, and the determination of intermediate poiats between I and P only. Et~~ .• - ~"1.'~" '~ •• r.

• I. 1"1.0". A41 •••

---,..------0 •• 29' C.r ..

.ALe.LAYt •• IF •• £ •• R. RICY' •• FOR P~." ,.

I + .0064 = .2 .. 4 .0380 6.4

I = .0360 x 2 .. 4 6.4

.OOQ4

I = .0135 - .0064 I = , •••• ,,' •• 4 te •••• r •• 4

.J.~&tl, •••

E'.~ • • 954.124. Arr •• ,.d • •• 954.131. AdJ •• t.d

---II1II--"--+ •••• ,. Cere

Elev P = 928.5520 ObserTed - 0 .. 0071 Cor.

Elev P = ~28.5449 Adjusted.

.... . -., . . . . .. e · . ....... ..... .....

.. .. • .. •

t-t

• •• .... .. (I

••

poiat at distance d. . .......

Correetion e. for any interil8aiate " from P aDd betweeD pl!>ililts I and P.

DEI '.'AI"" .F A& •• 8'IEI' FIRI'LA •

• ,0225 4.0 ',,' "",.",... 0 b C

8

- .00'71 = d

•. 0225 d C = ..:...;..----- +

~ 4.0 .0071 t. I

• £ R •

"l '1""4 -.. .. ...

COl .. •

Ii

.. .. ... • •

A D .. ' • (Wi) '1""4 • • • or

•

• o

., .qt 'I""'f

1C\ .. • ..

() . .....

E •

":'66-

I

•

L " • E to ~ 'I""'f ....

IC\ e ... -• • fit fit

POl ... e ' .. .e

• '""'" .... e • • ...

Thea C=(-.®05625d-.~"1)

.. e o • e

~ • IC\

• •

.. .... • • • •

e e .• •

I. st.,Jam.es,6.0 m.iles north of,on Republic road,6afe,2.5 miles east of,Miles 6choOl,25 feet west of, S.W.Cor.6.E.Q.tr.of-- the ~.W.~tr.,Sec.20,T 39 N.,R.6 W., 60 feet southeast of,concrete well curb, top of;bronze tablet. 1059.528 -0.030. 1059.4:98

12 Miles 6chool,0.75 miles north of, on Republic road,W.L. Martints house,150 feet southeast of,on top of hill,on east side of road, 12" post oak tree, in base of; copper nail and washer. ---~----- 1023.857 -0.025 1023.832

13 Miles 8choo1,1.2 miles north of, on Republic road,Bourbeuse River, 0.75 miles south of ford,at dim cross road,southeast corner of; painted square on sandstone boUlder. ------------------------ 924.251 -0.023

14 Miles School ,1. 75 miles north of, on Republic road,Bourbeuse River, 0.2 miles south of, fa..rm house, 600 f ·eet east of, on east side of road, lBn waJ.nut tree,in base of;copper nail snd Washer. ---~------------ 873.567 -0.020

"}:0 Wash School House,0.75 miles south of,Bourbeuse river ford,lOO ft.S. of,east bank of river,30" sycamore tree,in base of;copper nail and

924.228

873.547

washer. ------------------------- 839.998 -0.017 '839.981

T5a Bourbeuse River,at ford on Republic road, center of creek;road elevation. Road Elevation. -------------- ..... -- 834.000 -0.017 833.985

16 Bourbeuse river,0.25 miles north of, on Republic road,at T road east,N.E. corner of,in base of fence post; copper nail and washer. --------- 840.106 .0.016

l~ Bourbeuse Riv.,0.75 miles N.of,Wash 6ch.,400 ft.S.W.of,J.W.Crossnerts house,300 ft.S.E. of,B ft.W.of three mail boxes,on W.side of road,14" scaley bark hickory tree,in root of; copper nail and washer. --.. ------ 947.596 -0.015

-6~-

840.090

947.583

Adjustment of intermediate bench marks between the points P and Q on the Republic Road near the Phelps-Gasconade County Line •

• ~ w~" b. r •••• b.r.d that th ••••• P to Q I. a .pur "n. of •••• ,. 1ft oar .oh ••• of adju.t •• nt. Th. I'n. P to 0 .ae run a. a parafl •• dupllo.t. lfn. of ' ••• '.~a.d tb •• o.t probabl. yal ••• for tb •• '.Yatlo •• of the po'nt. on .uoh a II •• I. the ay.ra,. of tho •• Y.~al dat.r.lnct'one. Th. dra_ src. bal0. abo •• Ifne. of •••• ,. run by Brandt. and Qur'.y I •• tru •• nta:a. tb ••• Inatru •• nt ••• r. of .qual r.ff •••• t, .qua ••• 'Ibt ••• applJad to tb. work dona by .aoh In.tru.ant. Th. Ir ••• Itna r.pr ••• nt. tb. ayerasa cf tb. Inatru •• nt~l d.t.r.f •• tlon •• In tbe I.y.a ft.t .dJu.t ••• t pr.o •• dlftl, It ••• fouad tb.t the pofat P ••• 0.0071 ft~ hl.h. Ae the .pur lIn. p to Q bfn ••• dlr.otly on tb. polntP,tb.n the wbot. lin. p to 0 .fl ...... to b. Icwer.d 0.0071 ft.'ft tbo dfafr •• tho adJu.ted •••• I •• h •• n In blaok.

E •• y. P • 928_5520 Ob •• ryed • p. 928.5449 AdJust.d

- 0.0071 Cor •.

Correction c for any intermediate point between P and Q: C = average observed elev minus 0.0071.

T I

e O 0 ,..,

t " 0 .,0 0

()'\)

o <0

o o

o

Elo •• o • 997.2400 Ob.aryad • o· 997.2329 AdJ.et.d

- 0.0071 Cor.

~.,

o IX>

o o

o

,. " " . " ' . .,

J1 I' ,.

Go .... " 1'

,~

• I

1 "- A V E R : a E I

J r

o B Sr E R V E 0 .. I , L I It E .

p

... .... o o • o

• .0 -0 ... ::'0

0;; It.,,. '. "., (f .

IC'\

'" .• o

0 t<,. L 0 0 0 • E

R I

A D .I l:I S T

•

I ' ... 0 I"- (0

0 0 0 0 . . 0 0

! I . I

L o W E R

L. Nt; ~...I'.----

Gurley 0 r.a • •

I ... 0 ... <0 o <:> o 0

o co

C\ I"-

• q

Ie' N 0 0

t IC'

N 0 0

• 0

• ... 8RAPH FOR THE ADoJUSTUEIT OF .IUERIEDI' AfE POJIT8 8ETWEEN P AltII

-68-

. -

o.

18, Wash School,0.5 miles north of, on Republic road,Little Bourbeuse Creek, 8.t ford, on south banJ.c,on west side of road,lO" sca1 ey bark hickory tree, in base of; copper nail cmd washer. -- -- - 835.927 -0. (l)l0

P. Wash 8chool,1.0 miles north of, Little Bourbeuse Creek,0.5 miles north of,Harrison Schoo1,0.5 miles south of , Phelps-Gasconade County Line ,1.0mile south of ,Austin E. Bruer's house,200 feet west of, at T road east from Republic roe.d, on west side·of road,2 feet inside of fence,lO" post oak tree, in lJa.se of; copper nail and washer. --------------------- - -- 928.552 -0.007

19 Wash Schoo1,1.25 north of,Litt1e Bourbeu se Creek ,0.75 miles north o~,Harrison Sc11.00l,0.25 miles sOuXh of,Phelps-Gasconade County Line,0.75 miles south of,Steve Lort's house, 70 feet north of,T road west(Mail road to High Gate), nortr.!.west corner of roa.ds,24" oak stunlP in fence corner, top of; ten penny nail. ----------------- 9£2.732 Bra.ndis,

962.726 Gurley. - ' .. -----

Average 962.729 -0.007

20 Harrison School Ho use, on st. Jame sRepublic road,west side of School BUilding,north concrete door step, -top of; chiseled square. -------- 9£ '5."674 Brandis,

965. 658 Gurley. _ ... --_ .... --.. Average 965.666 -0.007

21 Harrison School House,0.25 miles north of,on Republic road,PhelpsGasconade County Line, 0.25 miles south of ,private--·T road west, south west corner of,16" blazed black oak tree ;bent nail in base of. -- 959.594 Brandis,

959.578 Gurley. _ ' ___ ' __ "iiir

Average 959.586 -0.007

-69-

835.917

928.545

962.722

965.659

959.579

Q. ira.rrison School Ho use,0.5 miles north of,on Republic road, intersection of st. James-Republic road and High Gate-Red Bird ro ad, 0.5 miles south of,Phelps-Gasconade County Line,28 feet south of,center of road,30 f eet west of, 5 feet inside wire fence, in line with center of road north into Gasconade County,dim T road west, 50 feet south of;iron post stamped "997". ------------------

P.B .. Me 997.242 Brandis, 997.237 Gurley.

Average 997.240 -0.007 997.223

-70-

Adjustment of intermediate temporary bench marks between point I at Miles School of Republic Road and Point J two miles due east on the Red Bird Road.

DER~.ATJON OF AOdUSTMENT FORMULA. El.~.I.IO'1.92~e Ob •• rYe

• l.lG'1 •• 9~4 A4'u.t. Ef.Y.d.968~394& Ob •• r •• • d.96a~3704 AdJu.t.

- 0 .. 0236 Cor.

Correction c. for any intermediate point at distance d. from J and between I and J:

~

: c

r----- - . ...

CI

I • • '!II -~ ... • ::'i'f

C - .0236 d =

.0060 1.6

C .0060 d

.0236 = +

.... 0

C =

C =

· I L o wE R •

1.6

( .00375 d

( .00375 d

E R V E 0

co \0 C'oI o • •

A D ~ 0 8 TED

32 33

~ 1ft

~ ~

-i • rot 0

+ .0236

~ .®236

_L I N E .

: C'oI 0

• 0

L Ii It E . 34

0 ~ •

411

)

)

'~AP. Foa T"~ ~D~uaTIE'T OF .tln EIUEB . ,U : E Pt.ITS

-71-

\0 on (\I

0

• 0

J

0 0

• 0

BEtweel I lKO - d4

Red Bird ~uadrangle. (Latitude 38 -38 15' ;longitude 91 30 1 -91 45 1 )

(Dawson Township)

Li ne f:co!il Miles School House on Republic Road, east via l'oad 2.0 miles, to point on Red Bird Road, two miles north of St.JD~es-Dawson Township Line.

I. 8t.JarJ.es,6.0 miles north of,on Republic Road, Safe,2.5 miles east of, cross roads at Miles 8chool,northeast corner of school house, s and s tone wheel guard,top of;painted square.--- 1061.925 -0.030 1061.895

32 Miles 8chool,0.25 miles east of, : on north and south center line . and between sections 20 and 29, R.6 W.,T.39 N., at di m T road north, center of road;road eleva 1052.867 -0.029 1054.838

33 Miles School, 0.75 miles east of,Rinehart's house,200 feet north of,at cross roads,southwest corner of,sandstone boul-ber, top of;dob o~. paint. ------ 1045.151 -0.027 1045.124

34 Miles School, 1.2 miles ea st of J

via crooked road through woods, Red Bird Road,O.4 miles west of, on south side of road,1 6 1f post oak tree,in root of;copper nail and washer. ------------------- 1037.012 -0.025 1036.987

J. Robinsons Branch, 2.5 mil es north of,St.James-Dawson Townsh ip Line, 30 m.iles north of, on top of hill J

Red Bird Road at T road west to Miles School,northwest corner of roads,15" white oak tree,in root of;copper nail and washer. ---- 968.394 -0.0~4

-7·'2-

968.370

Adjustment of intermediate temporary bench marks between point B at the intersection of Ozark Highway and Red Bird Road, and point H at the intersection of Jakes Prairie and Red Bird Roads.

DERIVATION OF A D JUS T M EN T

E'ev.B • 1074.6660 Ob.er.

FOR U U LA.

• B· 1074.6352 Adjus.

0.030 '8 Cor.

I\)

J, ~ . " ,CI] o .

<.) --- - - r t B ~ __ d __ >1 H

~----'---3.4:-----> ,

Elev.H = 1042.0580 eb •• r. • H. 1041.8535 Adju ••

0.2045 Cor.

Correction c. for any intermediate point at distance d. from B and between Band H.

and

simplifying:

c = -(0.051088 d + 0.0308)

and finally

c = -0.051088

co o 1("1

o

B

• eo -0

lIIO

'" :;; 0 .

35

0 ta

• co

d

C\

A

- 0.03(8).

'I ~

~ 'i'-

S '0

0 0 co

IC\ '0 ... 1("1 Q\ " ... co <> . C> • •

D J U S T 36 37 38

0 ta 0 .0 '" In

• • . ... ... ...

~

E 0

c = 0.173? d + 0.0308 ---3:-4---

" "

J ~ · '" \

\. L 0

• E ~

0 R ... IC"t ~ 0

<> ... GO . C\I

La 0- ..... ... . K'I '" ... . 1("1 ... . ... · .

L I N E . 39 40 41 42 H

0 0 IC\ 0 0

<> ca ..... 0 ... • •

C10I C10I (\I "" K"I

8RAPMFOR THE ADJUSTMENT OF INTERIlEO-IAtE POIIiTS ' BETWEEN B

-73-

A NO H.

35 '

36

37

38

Red Bird QMadrangle. (Latitude 38 -~8 l5 1 ;longitude 91 30' -91 45 1 )

(St. James Township)

Line from intersection of Ozark Highway and Red Bird Road,which is 1.0 mile northeast of st.James, thence north via Red Bird Road,to Royal F.O.,and Phelps-Gasconade County Line.

st.James,l.O mile northeast of, ' on Ozark Higbway,T road east, {intersection Ozark Highway and Red Bird Road),400 feet south of, Dr.E.A.Scottts house,300 feet west of,on west side of road,lO" elm tree, in root of;copper nail and washer. -------------------

st.James, 1.5 miles north of, intersection of Red Bird road and Ozark Highway, O. 5 miles north of,N.E.Johnson's house, 75 feet eabt of,on east side of road,20 feet . inside of fence,12" hickory tree,in root of;copper nail and washer. ----

Flag Spring Scho.o1, 0.25 miles east of, Britton's Store,0.25 miles west of, on Red Bird road, turn in road to east,30 feet south of, sandstone culvert across road,west end Wall,top of;chiseled square. -----------

Flag Spring School,0.5 miles east of,Britton's Store,in front of, center of road at turn to north;road elevation.--

Flag Spring School,0.75 miles northeast of,St.James,2.5 miles north of,at Y road to north,in center of Y,lO" post oak tree, ,in base of; copper nail and

1074.666

1069.326

1029.621 .

1039.320

1023.293

-0.031 1074.635

-0.057 1069.269

-0.084 1029.537

-0.096 1039.224

-0.118 1023.175

39 Flag Spring SchoOl, 1.25 mile SCi

northea.st of, Britton!l::.' Store, O.75 :-mi1es north of, dim. cross roads, southeast corner of, 10" post oa.k tree, in base 0:8.; copper nail and w.asher. --------- 1016.'~ 547 -0.134 1016.413

40 Brittons Store,l.25 miles : north of; st. James 3.5 miles north of) on': Red Bird Road, at ford across Robinson's Branch, 50 f:eet south of;projecting . Cotton Rock Sta.ne ·:; in bluff, on west side ;~' of road; chiseled square.--------------- 952.626 -0.159 952.467 (Bench Mark obliterated by dyna.~iteJ

41 Robinson t s Bra.'1ch, 025 mil e s north of, on Red Bird Road, at dim cross roads,tile culvert on .·~ we st side of road, south end waJ.l, top of; chiseled squa,re.---------1025.576 -0.172 1025.404

(Dawson Township)

42 Robinso,n's Branch, 0.5 miles

H' •

H.

north of, on Red Bird' Road, on st. James-Dawson Township Line, on east side of. ·- road, opposite T.-Snow's Mail Box,#12A,gate post, inc'hue of; copper nail and washer.-------- - ---------------- 1054.905 -0.185 1054.720

Robinson's Brru1ch, 0.9 miles north of, on Red Bird Road,at intersection of Rea.: Bird and· Jakes :Prairie roads;(Y road to northeast),F.Berries house, 300 feet south of, southeas,t c.~rner ~ of roads'i14" po sto~ tree, root of; c.opper nail and washer;. ----- -1042.947 -0.205 1042.742

Robinson~s Branch, O~ miles .. north of J on ,Red Bird r oad, at intersection of Red Bird and Jakes Prairie roads, (J road northeast) J F.Berries house, 230 feet south of; crotch in'.Y roads, opposite east end of "agon gate, five feet south of P.B~M. fence; iron poststamped "1042 "----1042. 058 -0.205 1041.853

Adjustment of intermediate temporary bench marks between point E at intersection of Jakes Prairie and Red Bird Roads, and point . J on Red Bird Road two miles nortb of St'i;,James-Dawson Township Line.

])ERIP'ATION OF AlJ){fSTMENT FORAllJLA. &b •• rv. ~.= .. ~ .- f E I • " • H. 104 2 • 05 II 0 j - .= .:=,,=

• H-l&41.853' AdJ •• t·N ~

--------- c ~ Cor. -- 0.204, ~ 0\

H~ _~ lJ ....... ..... - /.6' >

E1.v.J - 968.5980 Ob •• rv. • J - 968.3704 AdJuot.

- 0.2276 Cor.

Correction c. for and between Hand

any ipt~rmediate point at distance d.from H J.

and

TQen:

C = -(O.0144~?5 d + 0.2045) and C = (-O.01443?5 d

~ - .. ....

CI:'I .... 0 eN

•

H

• •• '-0 ... " ••

B S .E R V E 0 0 ... .

K'I ... ... N

A 8 JUS TEO 43

o C\

• o

L I N E .

: ... CII

liN e • 44

o ... .; ...

1 l 0 W e R

.. ..

\0 ,.. e.a N

J

o \0

• ...

0.2045) •

Adjustment of intermediate bench marks between point J on Red Bird Road two miles north of St :·James- Dawson Twp .Line ,and point M 500 feet north of Royal Post Office.

The adjustment of these intermediate benoh Marks can be effected by scaling the adjustment corrections from tme large diagram below and subtracting eaoh correction from the observ-ed ilevati0n as it + _ appears in the level notes;or~if desired,the adjust- . - ment oorrections can be com-puted by the foll- i _ _ _ _ _ owing formula for this section,Jto M. ~

I\:)

~ I' I

J.J, ... ...... -- -3.2 / _ ..... -"; ,j, M DERIVATION OF ADJUSTMENT FORMULA.

It.v.d- 968.598 O~ •• rv.d. • J- 968.3704AdJ.at.d. E'ev.U- 954.9830 O~a.ry.d.

• u. 9~4.7304 AdJuat.d. - 0.227' Cor. - 0.1726 Cor.

Correction c. for .any intermediate point at distance d.from M and between the points J and M.

- 0 B S

O - .0550 d + ln26 - -8:21-- .'(

C = (0.017134 d + 0.1726)

I r: R V Finally E 0 0=(-0.017 L , 134 d -0.1726 }

N E .

I ;.~-.

L 0

• : E (\I r- IC\ N L

: R ... r- 0- ... CD 0 . ... co 0- 0- 0 • IC\ \0 . 01 (\I 1"4 ... 0- r- ei

• . . • ... E r- .-• R ... ... . • .

J A 0 J U S T E & L t N E •.

M 4,15 ,46 .4'1 48 4~ 5. 51

<10 co 0 0 • IC'I pC • 0 ...

0- '" . eo ... • ·01 .... 0 IN . • • • • • - . . N N'I

-0 co ... pC pC 01 '" ::II

eRAPH FeR THE ADJU8TIIEIIT OF I NTERMEO<f,lTE POtllT8 BEf.EEI oJ AMD M.

-7~-

43 Robinson's Eranch, 1.4 miles I

north of,Township line,O~ miles nort}1 of, on Red Eird road, on top of hill, at T road west, north-west corner of, 12" white oak tree base of; copper nail and washer.---------~------------- 1036.078 -0.212

44 Robinson's Eranch,2.0 miles north of, Dawson Township Line 1.5 miles north of, on Red Eird Road, Eikman's house, 50 feet east of, on west side of road at gate, s~~dstone boulder, top of; chiseled square.------------ 996.226 -0.220

J. Robinson's Eranch 2.5 miles north of, Dawson st.James Townshi~ Line, 2.0 miles north of, on top of hill, at T road west to Mile School, northwest corner of ~oads,15" white oak' tree,root of; copper nail and washer ----- 968.598 -0.228

45 Robinson's Eranmh, 2.75 miles north of, on Red Bird Road, on top of hill, N.D.Putnamls house, 40" southeast of, on west side of road, five feet inside of yard fence, 12" waJ.nut tree, root of; copper nail and washer.--------- 967.970 -0.224

46 Robinson's Eranch, 3.5 miles north of, intersection of Red Eird and Jakes Prairie Road, 2.6 miles north of, ford at Clearcreek, aa miles south of, on Red Eird Road, s~~dstone boUlder at side of road, top of painted square. ----------------------- .. - 975.188 -0.211

47 Clear Creek, O. 5 miles south of, on Red Eird Road,intersection of Red Eird Road with Jakes prairie Road, 2.8:_.miles north of, 18 "Black Oak treeat side of road, in T1)O t of; copper nail and washeJ:S94.860 -0.208

1035.866

996.006

968.370

967.746

974.977

894.652

Adjnstment of point N on Red Bird Roa.d 0.9 miles north of Royal P.O., and point (;) on Jakes Prairie road 0.25 miles south of Phelps-Gasconade County Line. Nand 0 are intermediate JUDction points on the line II via Nand () to G.

M Q

via N So ' to G.

'" c-I .... .... . 0:>

M • .0

- 0 - • :::110

N Elev.M =

-~.-. -----------

- 8. '5"-

o .... --_ .. _--...,. Jr

G Elev. M = 954.9030 Observe

n M = ~§~~Z§Q~ Adjust.

. Elev.N = - 0.1726 Cor.at M.

Elev.O =

Elev.G = Elev.G =

;t:-

N 0 0-. C>

, Q oJ \~, C . ... <::::. ": '... .. Izet Y = Cor.to O;. then: C> " '

Let x = oor.to N,then: x - .1'726 _ 0.9 --O~0690- - 8:~5-

x= LQ§2Q_~_QL~ + .1726 8.05 '

Y - .1'726 _ 3.3 --O~0790- - 8~05-

Y = Lg§~~;~53~2 + .1726

Y = -(.0283 +.1726)

x= -(.OQ'77:1" .1726)

Then:

Then: -: : Y = - 0.2009 Corl to : o o. to

0:> eD o ... . 0

-0.1803 Cor.to 0 N. Elev.O=969.7360 Arranged x =

Elev.N=875.0710 Arrang d :-_~:;1803 Cor.

=.Q~g~Q§! Cor. w 0=969.5351 Adjusted.

,...,. r--~- ~

n N=864.890'7 Adjustu ========

A 0 JUS T E & o K'I . K'I

==== == ==

l , N E • G

'" co . eo

IRAPHFOR THE ADJUSTMENT OF 'NTERMeDIATE JUNCTION POINTS N

AND 6.

-79-

Adjustment of intermediate bench marks between point M 500 feet north of Royal Post Office, and point N 11(}O feet south of Red Bird Road Ford across Big Bourbeuse River.

The adjustment of these intermediate bench marks can be effected by scaling the adjustment corrections from the large diagram below and subtract1ng each correction from the observed elevation as it ap~ears in the level notes· or, if desired, the adjustment correct1ons can be computed by the following formula for this section of the net,M to N.

DERIVATION OF ADJUSTMENT FORMULA.

---~ .. j ...-''''' .

~ - -- -<0

<.u

~~~:::-...! lN EI ••• U • 954.9030 Db •• rv.

• M. 9~4.7304 ~djuit-Eltv. N • 87~.2;OO Db.trv.

• N. 874.8907 AdJu.t.

- 0' .1726 Cor. ,

Correction c. for any intermediate point at distance d~ from M and between points M and N.

and c = Q~I§§7_~ + 0.1726 u.Q

c = -(0.20744 d + 0.1726) or

... .....

>0 r-c-a C\I r- • ... 0

• o

A D J U S T E52 M • eo 0--0 ..... -. • :EO 0

C = (-0. 20744 d

<:)

l t N E • N

. o

GRAPH FOR THE ADJUSTMENT OF INTERMEDIATE POINTS BETWEEN M AND N.

-sc-

0.1 '726).

48 Royal Post Office, 1.5 miles south Of, on Red Bird Road, Clear Creek,bridge over,south east end of . concrete brid~e, elevation on hand rail.------ 872.690 -0.200

49 Royal Post Office, 1.2 miles south of, Clear Creek bridge, 0. ;:, n iles north of, T road east, lO(!1" south of, concrete culvert across road, center of; road elevation. ------------------- 924.914 -0.194

50 Royal Post Office, 1.0 mile south of, at T road northwest, 24" Black Oak tree on west side of road, in base of; copper nail and washer.------- 983.208 -0.191

51 Royal Post Office, 0.25 miles south of, on Red Bird Road,on east side of road, 30" White Oak tree, in root of; copper nail and washer.------- 985.210 -0.178

Mt Royal Post Office, on Red Bird Road,Steen's Store, north end of concrete porch, one foot out from Building, six inches from north end of porch;pronze P. B.M. tablet stamped "961!' ------ .. --- 960.631 -0.173

M. Roy-ail Post Office, 500 feet north of,Royal School House,lOOO feet north of,at T road west to Republic Road,soutPwest corner of, telephone pole,in base of; copper nail and washer.------- 954.903 -0.173

52 Royal Pqs.:-t Office,(l.6 miles north of,on Red Bird Road,Huntsman's House, 200 feet north west of,on east side of road, 16" Red Oak Tree, in root of; copper nail and washer.--------935.969 -0.274

-81-

872.490

924.720

983.017

985.032

960.458

954.730

935.695

Adjustment of intermediate bench marks betweel'l the points Nand R OD. the Red Bird Road near the Phelps-Gasconade County Line.

!I

l o IV E R

I • . !IEI .. V~ .N.87~. 2500 Obser.

• ~ . , . N.874.8907 Adjus. ! I --------~ I - 0.3593 Cor.

)IE.e".RaS66.3380 Obser. ~I • R-s65.9787 "djus.

~I :-'0:;;;3 Cor.

j :correction c. fer any ,Iintermediate lJoint ' Ibetween N and R~ is:

!II ()

8 r a II d .. '.

OBSERV E D LINE.

... ~ I~. G" r 1., \ ~ 1'1

l.V., Inetru. t

II •

I N to R i8 a spur line of I levels in the level net l I ad.justment soheme. The line ~

N to R was run as a parallel duplioate line of lev- E els,then the most pr0bable R values for the elevations • of the points on such a line is I the avera~e of the several deter- I ~ minations;The diagram below sbo.s : lines of levele run by Gurley and iL

I. Brandie Instruments; as these CrJ ~ instruments were of equal re- I ~ I~ finement,equal weight was app- I ~ ~ lied to the work done by eaoh I '

I ~ instrument. The green line rep- I I resents the averag e of the ins-

I I Ie = averaged observed

i I elevatio~ minus 0.35931 : 'I

trumental determinations. In the level net adjustment preceedinR., it was found that the point N was 0.3593 feet high. As the spur line N to R hinges direotly on the point N,then the whole line, N to R will have to be lowered 0.3593 feet. In the dia~ram, the adjusted line of levels ~ is shown below in blaok • , I

. 1

! I I

i I ! I

feet. I

I I I I I I

~~' -= -:-=' -'::::::-=-~~' .:::: ' ..:.. . . - . - ~~ - - - - - - - - - - - - - -- __ I~ ~-- .. -.---.. -.--... - .... - .. _.-_ .... _.- 0.5· '-'·'·"-" ..... . ..... -..... --. -..... -. - ... ----... - .. ...........)-~ A 0 JUS TED lIN E • R

' ~3 GRAPH FOR THE ADJUSTMENT OF INTERMEDIATE POINTS BETIEEN NAND R.

-82-

N. ~oyal Post Office,1.25 miles north of,on Red Bird Road, at T road east to Jakes Prairie Road, 50 u feet south of T road,on east side of road,14" Red Oak Tree, in base 'of ; copper nail and washer.-------------------------875.250

53 Royal Post Office,1.45 miles north of,on Red Bird Road,Ford across Big Bourbuese River,125 feet southwest of,center of road,30 feet west of,main channel of River,80 feet so uth of,leening 12" slippery Elm Tree in south bank of River,inbase of; ten penny nail.------~------------- 812.023

R: Royal Post Office,1.75 miles north of,on Red Bird Road,Red Bird Post Office,2.00 miles south of,intersection Red Bird - st. James and Red Bird - High Gate roads,1.50 miles south of,Ford across Big Bourbuese River,O.3 miles north of, large Red Barn on east side of road,lOO feet southwest of,center of road, 25 feet west of, (Primary Tr averse Station#87),wor.m fence,4 feet inside of,in lot;Iron Post set on Phelps Gasconade County Line ·, P.B. M. stamped "866!'-------------------866.338

-83-

-0.359 874.891

-0.359 811.664

-0.359 865.979

'Adjustment of intermediate bench marks between point P on Republic Road 0.5 miles south of Harrison School, and point M on Red Bird Road 500 feet north of Royal Post Office.


EI·y.p·928.7~10 o •• tl-:':":"':'- - "':}- -'----- j E'·.,.t.I-954.9030 0 ••• • P-g28.-5449 Ad", t • "-954.7304 Adj. -------- ~ . ~ --- -----

Cor. . ' - O. 1 8 6 1" c +---d ~ I\) Cor.. - O. 1 72 6

I i p"f 'M ~- -- 2,4-= ------

Correction c. for any intermediate point at distance d.from M and between P and I.

.... \0 CD .... •

0

• •• -0 -. :II.

GRAPH 'QR THE

-l 0 W E R

C - 0.1726 = __ ~ __ --~:5I§~---- 2.4

C = -(0.005625 d + 0.1726)

and finally

C = (-0.005625 d 0.1726).

o B S E R V E 0

... eN co .... •

0

A D J a S TED 54

IC\ ,.. .. co

l- J N E •

'" co ... eo ,.. ,.. .... .... . • co 0

L " , 55 Ijb

• IS .... ... • i ... ...

E •

co '0 ,.. .... co

0 O't

• ... Ae.tUSTI£NT OF ""TER8ED1,.,'E PfUtTS

-84-

-

'0 C'I

L ,.. 0 .... • co E R

I

0 .... •

eN

8EtIE,EIt P AND M.

Red Bird ~uadrangle. (Latitude 38 -3815 1 ;10ngitude 91 30'-91 45')

(Dawson To'wl1ship)

Line from paint on Red Bird Road 500 feet north of Royal Post Office, thence west 2.5 lliiles via road to point on Republic Road 1.0 mile north of Wash Schaal.

M Royal Post Office,500 feet north of ,Royal School House ,1000 feet north of,at T road west to Republic Road, southwest corner of, telephone pOle,in base of;cop-per nail and washer. ------------ 954.903 -0.173 954.730

57 Royal Post Office,0.5 miles west of,at foot of hil1,at private road southwest to farm 11ouse,011 north side of road,infence corner,twin white oak tree,in base of;copper nail and washer. ----- 863.080 -0.176 862.904

56 Bourbeuse River Ford I at mouth of Little Bourbeuse,middle of f ord; road elevation. ---------------- 813.000 -0.178 812.822

55 Royal Post Office, Ito- west of, S.T. Mitchel's house,GOO feet east of,Bourbeuse River ford, 700 feet west of,on north side of road,at corner of woods,20n white oak tree,in root of;copper nail and washer. ----------- ---- 831.792 -0.179 831.613

54 Royal Post Office,1.65 miles west of,Bourbeuse River ford, 0.75 miles west of,Republic Rd., 0.75 miles east of,Phelps-Gasconade County Line,l.O mile south of, south side of road,at T road south,12 n white oak tree,in base of;copper nail and washer. ----- 924~692 -0.182 924.510

-B9-

P. Wash School,l.O mile north of, Little Bourbeuse Creek,0.5 miles north of,Harrison School, 0.5 miles south of,Phelps-Gasconade County Line,l.O mile south of,Austin E.Breur's house 200 feet west of,at T road east from Republic Road,on west side of road,2 feet inside of fence, 10" post oclc tree,in base of; copper nail ~,d washer.-------- 928.731 -0.186

-86- ·

928.545

Adjustment of intermediate beneh marks between point E at Knobview Depot~and point D at the intersection ef tbe Ozark Highway and the Phelps-Crawford Line.

Cor. •


0.000

E

Adjuet.

ft. Et.v~D· 1098.8499 Observe

• D- 1098.8847 Adjust. ---------to 0.0357 Cor.

Correction c. for any intermediate point at distanee d.from E, and between the points E and D.

• .0 -0 - .

C = d a:-035?- -1::-(5-

Then:-

C = (+0.0357 d). o o

E~~ ______ ~A~D~J~~U~S~T~E~D~ __ ~L~I~N~E~~.----~n

I R o -.... A

I o S

E

( I 1 II

2'

.. - !)

GRAPH FOR THE ADJUSTMENT OF INTERMEDtATE POINTS BETWEEN E AND D.

c

'" ..., o • o

Adjustment of iDtermediate beDch marks between point D at the intersection of the Ozark Hi!!hway and the Phelps-Crawford County Line,and point G at the intersection of Jakes Prairie and Tribby roads,O.5 miles south of Hinkle P.O. .

DERIVATION OF AO~USTKENT FORMULA. Ele".O-1098.849 Oh.er.

• 0-1&ge.8847Adj •• t. E le".6-921.2020 Ob.er".

• G-921.2814 Adjust.

• O.0357Cor. + 0.0794 Cor.

D .(-- ----- 4-.$------- > G -~ ~

~ - -----d --.. --- - i ~ ~ 7 _ _____ _ __ c _ _ __ _ ~

. · r Correction c. for any intermediate poin't at distance ' d.from point D and between points D and G.

c=(+O.OOe?11 d + O.035?)

• eo 0 lr\ 0 IC\ IC\ IC'I 0 0 0 11:'10 -0 IC\ C\I IC\ ,... (\I -0 0 IC\ 0 c:.. lei . . • . =- 0 ... ... r-I (II (\I 1(\ 1(\ .... .... .... D 606162 63 64 65 66 6'7686

oJ u s T E D L I N E

R (\I (\I

-oA 0 0 -0 I

0 0 \C) \OS

D. • 0 °E

0I :: 7· 1 G. ,

GRAPH FOR 'HE AOJUStMENT OF INTERMEDIATE

POINTS BETWEEN 0 AND B.

-88-

Red Bird ~uadrangle. (Latitude 38-3815';longitude 9130'-9145')

(St.James Township)

Line from Knobview Depot,east 0.5 miles via Frisco Railro8~d to Phelps-Crawford County Line, thence north via Pheps-Crawford County Line road,3.5 miles to T road west, to Jakes Prairie road, thence west via T roe.d,1.5 miles to Jakes Prairie road at a point 0.5 miles south of Hinkle Po st Office.

E. Knobvie\l!}' Depot, 90 feet e8,st of, 45 feet south of center of track, 12 feet east of R.R. crossing, 2 feet north of R.R.Property line, iron po st st81n.ped "108411. ------ -- 1074.005 0.000 1074.005

58 Knobview Depot,O.5 miles east of, via R.R.,concrete serr~phore base, southwest corner, top of;signal 949 on Phelps-Crawford County lineipainted square. ------------ 1081.484 0.018 1081.502

D. Friendship School, 2.0 miles east of, on Ozark Highway,at cross roads on Phe1p s-Crawford County Line, SOt:.. thwest corner of road s, 14" hickory tree, in base ofjcopper nail and washer.1098.849 0.036 1098.885

59 Knobview Depot,1.5 miles north of, 011 Phelps-Crawford County line r02,d, Ozark Highway,0.5 miles north of, Cucaro10's house,75 feet northeast of,on west side of road,in fence 1ine,12" locust tree,in base of; copper nail and washer. --------- 1071.411 0.040 1071.451

(Dawson Township)

60 Knobview Depot,2.25 miles north of, on Phe1ps- Cr.awford County line road, Ozark Highway,1.25 miles north of, drain to northwest,lOO feet south of, on west side of road,20 n white oak tree,in base of; copper nail and washer. ------------------------- 1064.348 0.048 1064.396

-89-

61 ,Knobview Depot,2.5 miles north 'of, on Phelps-Crawford County Line Road,O::;ark Highway,1.5 miles north of,on top of hill, opposite nail box on east side of road, center of road;road elevation. ------------------- 1087.589

62 Knobview Depot,2.75 miles north of, on Phelps-Cravlford County Line Road,Ozark Highway,1.75 miles north of, fe.rm house, 350 feet northwest of,on rocl< projecting out of ground in center of road,15 fE.:et northwest of fence corner,top -of rock; painted square. --- ... ---------- 1012.794

63 Ozark Highway, 2. 25 miles north of, on Phelps-Crawford County Line Road,at dim cross roads, on top of hi1l,15 feet southwest of roa.ds,on stump; painted square. -------------- 1071.623

64 Ozark Higbway,2.65 miles north of,on Phelps-Crawford County Line Road,Hinkle P.O.,225 miles southeast of,on top of 11ill,in old abandoned field, west side of road,conglomerate boulder,top of;painted square. ----------- 1040.245

65 Ozark Highway,3.00 miles north of,on Phelps-Crawford County Line road,right aneled turn in road east,75 feet south of,T road west to Hinkle P.O.,just opposite,on east side of road, sandstone boulder,top of; painted square. -------------- 994.501

66 Phelps-Crawford County Line, 0.5 miles west of,on Tribby road,Hinkle P.O.,1.5 miles south east of,30 feet east of drain, on south sj.de of road, sandstone

LboUlder,top of;painted square. 994.905

-90-

0.050 1087.639

0.053 1012.847

0.057 1071.680

0.060 1040.305

0.065 994.566

67 Phelps-Crawford County Line, 1.0 mile west of,on Tribby road, Hinkle P.O.,l.O mile southeast of, at dim cro ss roads,lOfeet south of center of roads, send-stone boulder projecting out of ground,top of;painted square.-- 950.010

68 Phelps-Crawford County Line, 1.25 miles west of.on Tribby road , southeast corner, S. W. Q,tr. of·-· the S.E. Q,tr. Sec. 25, T. 39N., R.6 W.,at private road north, mail box on north side of road, center of road just opposite; road elevation. --------------- 949.773

G. Hinkle Post Office,Q.5 miles south of~ intersection Jal{es Prairie and Tribby roads,(Troad east)(sign board 10 miles to Cuba), east side of Jakes Prairie road,30 feet no~th of road intersection,8 f1 walnut tree, in base of;copper nail and washer. ----------------------- 921.202

-91-

0.074 950.084

0.076 949.849

0.079 921.281

~djustment of intermediate temporary bench marks between point A- at St ~James and point Bat the intersection of Red Bird Road and Ozark Highway.

OERIYATtON OF ADJUSTUENT FORMULA.

EI.y. A - 1081.800 Cor.- 0.000

i c

I

I .. () ., o <i

E

E' ••• 8 ,. 1074.66'0 o~ •• rY. . • B. l074.63~2 Adj._t.

0·. Q::5 0 8 Cor.

Correction c. for any intermediate point at distance d. from A and between points A and B.

c = .0308

"" = v

C =

a 1.0

.0308 d

1.0

0.0308 d.

'"" ./ ... ell 1"4 CIt

• CIt

A 0 J U -s ·T ' E ' 0 . .

9 ... •

CIt

:;"92-

/

/"

' L t. ' N E •

OCI CIt

K'I CIt . CIt

CIt CIt

• ...

Adjnstment of intermediate bench marks between point B at the intersection of Red Bird Road and the Ozark Hil!hway,and point C on the Ozark Highway at Friendship School.

The adjustment of these intermediate bench marks can be effected by scaling the adjustment corrections from the large diagram below and combining each correotion algebraioally w1th the observed elevation as 1t appears in the level field notes; or t ~t desired, the adjustment corrections for the intermediate p01nts can be computed by the following formula for this section of the level net,B to c.

DERIVATION OF ADJUSTMENT FORMULA. E'.y.S • 1074.6660 Observe Eley.e • 1083.4« 0 0 Ob •• rv.

• S· 1074.6352 Adjust. • e. 1083.4299 Adjust. -.-:_-----0.0308 Cor .. .. 0.0299 Cot ..

§--~--- ~t' -B gL S ---- - ",C I - ~. I --d---~::: '-'--- ~ ~--- - ---- --- -- ~ ; ~ 8.0----- ~

Correction c.for any intermediate point at distance d.from C and between the points Band C.

and O.Q299

c = -(Or020233 d - 0.0299) then C = (+0.0299 - 0.020233 d).

C> C>

C> <:> <> <:>

R B I - j

IA 0 .. U S T E t • I I 1 :1 ~ LINE. I l I +

c C\I A 0-

I 1 .1

,I I I

I i ' I I I ,

L __ _ _ , _ _ L _ _ _ __ _ _ , _ _ ..J. _ - - - -

13 69 '70 '71 • eo <> In 0

- 0 '" COol 0 ... • • • • ::IIIt(\ N c-I c-I

'72 '73

IC\

• pol

N

• pol

C> 0-

+ t. I "'e- ~ N <:>

I • I .. _ J ___ _ '--

'74 G C1

o c IC\

• o

o . C!>

GRAPH FOR THE ADJUSTMENT OF INTERME~ . ATE

PO.NTS BET.EEN B AHD C.

-93-

A.

22

B.

69

70

'71

M~RA.EC SPRINGS QUADRANGLE.

(LatItude ~7·45~ -~8·; lo.sftude 91·30~ -91-'5')

, .'

LINE FROM ST. JAMES EAST VlA OZARK KrGHIAY TO PHELPSCRA~FORO COUNTY LIHE# SAI8 POtNT ONE MILE NORTHEAST OF KNOB-vIew RAILROAD DEPOT.

Oe.crlptlon.

(See description on pag~ 64: )

St. James 0.4 miles east of, on St.James-Ouba road(Ozark Highwayl, Frisco R.R •. ,50ft. north of, turn in road to north, 500 ft. west of, concrete culvert, acrossroad,north end wall, top of;chiseled square. ----------

Observed ElevatIon

1e8'7.8oo

1065.363 RED BIRD QUADRANGLE.

COR.

0.000

-0.0124

(Latitude 38·-38°15~; longltude 91°30' -91 0 45')

St.James,1.0 miles N •. E. of, Ozark Highway, T road east (intersection of Ozark Highway and Red Bird road},400 ft. S.of, Dr.E.~.Scott's house, 3G8 ft. w. of, on W. side of road, 10 ft elm tree, in root of;copper nail and washer. -------------------- +074.666 St.James,1.5 miles ~.E. of, on Ozark Bighway,O.5 miles east of Red Bird road intersection, on ~ N.side ofroad,~2" post oak tree, in base of; copper nail and

-0.031

washer. ------------------------... - 1062.669 -0.C21 St. James,1.75 miles northeast of, on Ozark Hi~bway, concrete culvert aeross road, south end wall, top of; chiseled triangle. , St. -i •• es~, .9.2 miles N.E. of, on Ozark Higbway, on top of hill~ center of road; road elevation.

1034.437

1954.446

-94-

-0.015

-0. Ole

AdJu.ted Elevation

1087.800

1065.351

1074.635

1062.648

1034.422

1054.436

Adjustment af intermediate bench marks between point C on the Ozark Highway at Friendship School, and paint D at the intersection of the Ozark Hi~hway and the Phelps-Crawford County Line.

The adjustment of these intermediate temporary bench marks can be effected byscalind the adjustment correotions from the lar~e diagram below ana combinind each correction accordind to - its s1gn with the observed elevation as it appears in the level f1eld notes; or~if desired, the adjustment correotions oan be Gomputed by the following formula for this section of the lavel net,C to D.

E tev.C • C

DER.fAT A D JUS T • E T

o N 0 F

FOR M U LA •

• 1~83.4000 O~.erY.

· ;~!;~;;~t(:~'~~~ _ u~tJID ~- -- 2.2 S - .?

Eley.D • 10~i.9110 Ob.erv. • o. 1098.8847 Adjust.

------~--0.0263 Cor.

Correction c. for any intermediate point at distance d. from C and between the points C and D.

and

c = -(O.~24978 d - O~0299) then C = (+0.0299 - 0.024978 d).

I"o o o

L 1<"1

<:> ,\'" 0, \C \. \ CII

E <:> R •

r---T---~~~----~--------~~~~-I I R ADJUSTED I LINE. A I

0.. I 0.. l 0 I ~ S ~~~ I o E s~ I • /' ()~ I I • :.... -1 _______ .L _____ 1-,- __ _

Sl 75 76 77 D

• • 0 1£\ tc\ CC\ - co ON ... I"-... . . . • :::E 0 0 rot ...

9RAPH FOR THE ADJ~STWENT OF tN.ERMEDtATE

POINTS BETWEEN C ANa D.

-95-

73

74

st. James, 2.5 miles northeast of, on Ozark Righway, Friendship School, 1.5 miles west of,concrete culvert(errected 8-13-20 by T. RaIl), north end waIl, top of; chiseled square.--------------1010.05?

Friendship School, 1.25 miles west of, on OzarkBi~hway, at T. road N. to Binkle P.O., Corrugated iron culvert, rock on east end of culvert, top of; chiseled trian~le. ---------------------- 1018.926

FrieDdship School, 0.5 miles W. of, on Ozark BiQhway, Gerber's Store, 600 feet W. of, concrete culvert, north end wall, top of; chiseled trian~le. -------------- 1062.045

C. Friendship Bchool, on Ozark iji~hway, southwest cornerof concrete porch of school house t top of; chiseled trian~le, ~aiDtea "1083.3". -------------- 1083.400

G1- :Frienli "Semoolf "on .021ark Highway, !jOfeet S.W. of, in School Yard, 10 ft. from fence corner,30 ft. N. of center of Ozark Hi~hway, 30 ft. E. of center of T. road

75

76

north; iron post stamped "1E)83". l083.458

Friendship8chool, 0.25 miles E. of! private T. road N. to Louis Ball's residence, ~5 ft. W. of, concrete culvert, S. end wall top of; chiseledsguare.---------:J:037.041

Friendship Sebool, (!,.9miles E. of, on ozark Bi~hway, PhelpsCrawford County 'Line, 1.1 miles west of, concrete culvert, south end end.wall, top of; ehjseled S~u8re .. - .... ---.-~------... ------------1042.29'7

-96-

0.000 1010.057

+0.005 1018.931

+0.020 1062.065

+0.030 1083.430

+0.030 1083.488

+0.024 1037.065

+0.001 1042.298

77 Friendship School, 1.5 mile~ east of, on Ozark Highway, Phelps Crawford County Line, 6~~ milee ~ west of, Knowview, O.~ miles north of across field, concrete culvert under Ozark Highway, north end wall, top of; chiseled square.-----1028.412 -0.,014

D. Friendship School, 2.0 miles east of, aD Ozark Highway, at cross roads on Phelps - ~rawford county Line, southwest corner of roads, 14" bickory tree, in base of; copper .nail ana washer.--------------------------- 1098.911 -0.026

1028.398

1098.885

Th:,s cor.pletes the descr.?,ptHJ1I. and adJustment of the b~"~ oj levels from St. )1l1r.es east v?,a OzarkS"gb:ay to the Pile ~;:s.=C,.a:IR:j ord Count J' L; r.le~ 'sa:.d ,/>oHIi one lIn l e nort hefZ'st of K •• tv'/,ewR~llroad Depot.

-9~-

Adjnstmellt of intermediate beneh marks between paint e an the Ozark Hi2bway at Friendship School,and poist F on the Jakes Prairie Road 1.0 mile s cutii of Hinkle Post Offiee •

.3.0,-- .. · _._._- ' -' , - > F __ "~~ ' ____ f _____ I

c

~ER'VATION OF AOJUaTMENT FORMULA.

Et.v.e • 1083.4aoo Ob •• rv. • e. 1083.4299 Adjust. - ........ _-- ... -

+ Ih02'99 Cor.

Correction e. for any intermediate and between the points C and ,.

- .

Etev.F • loti.1770 eb~.rY. • F. 1001.3429 AdJuet.

---------+ 0.1659 Cor. point at distance d. from C

8RAPH FOR THe ADJUSTMENT ~F fNTERMeolATE POINTS BETIEEN C AND F •

• • 0 0 0 0 0 -0 IC'I IC'I C) 0 .- . • • • • 20 , 0 ... C'I on ~ ' . 78 79 89 F - , . ;.;. ~'

: ADdUSTED LI ME. • 0 001 IC'I R

" 0 0 A co 0 I 0\ 0

°Oe 0 CII s Algebraic eR" r-4 E 0\

eo IC'I

Solution. 100 ... C ,.. ~LQ299 = d .. -0: 360-- -3:-0- , I 0.1360 a c = + 0.0299 IN --

---~:~- - r: •

-.

(+0.04533 0 .. (299). " C = d' +

-98-

Red Bird ~uadrangle. (Latitude 38-38 l5';10r.gitude 91 30'-91 45')

( St. J <?me s Town ship)

Line fr.om. Friendship School on OZCI.rk Highway, in center of section 11,T. 38 N., R.6 W., north via road 3.0 miles to paint on St. James-Jakes Prairie road,l.O mile south of Hinkle Post Office,at a T road south east.

c. Friendship School, onOzark highway,in center of sec.ll,T.38 l'r., R.6 W.,southwest corner of concrete porch of School House,top of;chiseled triangle .. ---------- 1083.400 0.030 1083.430

C'.Friendship Schoo1,on Ozark Highway,in center of sec.ll,T.38 N., R.6 W. J 50 feet southwest of, in school yard,lO feet from fence corner,30 feet north of center of Ozark Highway,30 feet east of center of T road north;iron post P. B. M. stamped.--------------~-------- 1083.458 0.030 1083.488 "1083". .

78 Friendship Schoo1,O.5 miles north of,Hinkle P.O.,3,5 mi~~s south of, T road west,east side of road,8" black oak tree,in root of;copper nail arid Washer. -------------- 1018.238 0.052 1018.290

(Dawson Townshi,)

79 Friendship Schoo1,1.5 miles north of, Hinkle P. 0.,2.5 miles south of, on township line between St. James and Dawson Towr.ships,bridge on township line road over south fork of Clear Creek,northwest corner of bridge; co~per nail and w~asher. ------- 951.418

80 Friendship School,2 .. 0mi1es north of,Hinkle P.O.,2.0 miles south of, east side of road,inrock ledge; chiseled triangle.-------------- 945.415

-99-

0.098 951.516

0.120 945.535

F. Friendsh ip School, 3.0 miles north of,Hink1e P.O.,l.O mile southwest of,on .Jakes Prairie road,at T road southeast(road to Friendship Sch.) southeast corner of roada,15" white oak tree, in base of; copper nail ~~d washer.------ 1001.177

· -100-

0.166 1001.343

Adjustment of intermediate bench marks between point B at the iD.ersection of Red Bird a,nd .Jakes Prairie roads,and point F on Jakes Prairie Road 1.0 miles south ' ~ of Hinkle P.O. and 3.0 miles north of Friendship School.


E'ey.H • ID42.0580 ~b.erY.d. • H· 1041.8535 Adju.t.d.

E 'ey.F- 1801.6110 Oll •• r".d. • Fa 1001.3429 Adju.ted. -----_.-- ---------

0 .. 2045 Cor. 0.2681 Cor.

E:~~= - ~:=--~-=:f n ~ ~ ~J~ H ~ - I. 7.s----~ F

Correction c.for any intermediate point at distance d.from H and between the points Hand F.

a .. o

. C\I ' ..

H • HI .0 -0 ... . .0

c - O~2045 = d --5:0636-- -I:?5

C = -(0.036343 d + 0.2045)

C = (-0.036343 d

A 0 J U S T 81

IC'I

a

E 0

0.2045).

L o • E R

L I

L I

N E •

o o IC\ C\I

N E 82

IC\ eN ,. ...

.

GRAPH FOR tHe AD.I'IIS,"eWT OF 1I1TERME91A~E

P&UUS . BETWEEN 8 A Itt F.

-101-

• IC\ ... . 0'4

Red Bird ~uadrangle. (Latitude 38 -38 15 t ;longitude 91 30'-91 45 1 )

(Da,yson Tovmship)

Line fro~ intersection of Red Bird and Jakes Prairie roa.ds, thence 110rthet'.st 2~ 75 mile s via Jake s Prairie road to Collins School and Hin.1<:le P.O;, thence north 4.5 miles via JeJce s Prairie road to Phelps-Gasconade County Line, 2.0 miles west of Jakes Prairie Post Office.

H Robinsons Bral1ch,0.9 miles north of, on Red Bird Road,at intersection of Red Bird and Jakes Prairie roads, (Y road }Tortl:east), F .Berrie shouse, 230 feet south of,in crotch in Y roads,o~posite east end of wagon gate,5 feet south of fence;iron P.B.M. po st stamped "1042 fl. ----- •• -------- 1042.058 -0.205 1041.853

81 Intersection Red Bird and Jakes Prairie roads, 0.75 miles northeast of, on Jakes Prairie road,me.in fork of Clear Creek, O. 5 miles southwest of, via way of road, H.Eikr:laJl 1 e house,lOO feet east ofzon eBst side of' road, 14 It po st OC'l.Jc tree, in base of; copper nail and washer. ------;.. 964.540 -0.231

82 Intersection of Red Bird and Jakes Prairie road.s, 1.25 Llil es northea.st of,on Jakes Prairie road,Hinlc1e P.O., 1.5 miles southwest of,Clec.r Creck Ford, (Main fork), 50 feet east of, at angl e in road to north, southeast corner of road,in fence li#e.20"hickory treo, in base of; cOJ?:per nail and washer. ---------- 91G.09l -0.250

F Friendship School, 3.0 miles north of, Hinl<:leP.O.,1.0 mile southwest of on Jakes Prairie road,at T road southeast(road to Friendship Sch.), southeast corner of ' roads,15 "whi te oak tree,in base of;copper nail

964.309

915.841

and washer. ------------- ... -------- 1001.611 -0.268 1001.343

· .... 102-

Adjustment of intermediate temporary bench marks between point F on Jakes Prairie Road 1.0 Mile south of Hinkle P.O. and 3.0 miles north of Friendship School, and point G 0.5 miles south of Hinkle P. O. at the intersection of .Jakes Prairie and Tribby roads.

Deriva~ion of Formula for ~he Adjustmen~ of Intermediate Points.

Elev. F • 1001.6110 Ob.erv.d. Elev. G • 921.5450 Ob.erv.d. • F. 1001.3429 Adjus·ted. • Q. 921.2814 AdJustsd.

--.---~-~ 0.2681 Cor. - 0 .. 2636 Cor.

t I\J

~ ())

t .. -- ~ - 4' -::.-.::::,.:~--~ -I\) . I Il) .

~ C ---- -d ..

F ~ 1 . ~~---.-- ---~-o~.s--... -.. ---~~ G

o 0045 d C = -~O:5--- + 0.2636

C = -(0.009 d + 0.2636)

C = (-0.009 d - 0.2636)

F

o 8 S E R V EO . _.L .1..1'1 .. <:_ •

. ~ J=~!:e:~ ~. ;nL~::, ~~:t- ~ ': ~ level field notes show .w that no intermediate o~ points were establisbed ~

• in ~his sec~ion of the ne~.

A 0 JUS TEO liN E • G

0.00 MI. 0.50 III.

GRAPH SHOWING AD~USTIIENT OF PotNTS

F AND Q.

-103-

Adjustment of intermediate bencb marks between point G, 0.5 miles south of 'Hinkle P.O. at the intersection of Jakes Prairie and Tribby roads, and point () on Jakes Prairie Road 0.25 .iles south of the Phelps-Gasconade CountyLine.

The adjustment of these intermediate temporary benoh marks oan be effected by scaling the adjustment corrections from the large diagram below ana subtraeting each oorreotion from the observed elevation as it appears in the field level notes;or,if desiredtthe adjustment corrections can be 00.puted by the follow1ng formula for this section of the leyel net,G to O.

~f-' ~ - 1t~ __ ----j-~ Co c ---d.--- I\l t I ~

. ~ ' ~ . ~

G ~~.7S ) 0 DERIVATION OF ADJUSTMENT FORMULA.

Elev.G . 921.!H50 Ob •• ryed. E 'e.,. 0 • 969.1580 Ob •• rved. • G = 921.2814 Adjusted. • 0 . 969.5351 Adj.sted.

-------- --------- 0.2636 Cor. - 0.2229 Cor.

Correction c. for any intermediate point at distance d. froll and between G and o.

and

c = -(0.008568 d + 0.2229) or C = (-0.008568 d - 0.2229).

{J R ~ if. if V Ii' fl r--- -.-- _ . ~ . - L f ~. ~ .- - -- ._-. - ---.

I ' . .

0 C> 0 0 '" 0 C> '" «I • 01 • r- :;: '0 N

"" IC1 • • IC"\ L N Col N N N N N 0 CN N '" . . • . . w ~

. . co 0 0 co 0 E

0 0 o

R

A D J U S T E D l f N E . G 83 85 86 87 88 89 90 e • 8J: .'" '" '" co 0 0 co co

-r- C<I \0 0 K'I 0 0 Ie'I C> - . • • • . . :.. • IC"\ IC"\ N C<I .... 0 0

GRAPH FOR THE ADJgSTMENT Of INTERMEDIATE

POINTS BETWEEN G AIm O.

-104-

0

• G ~inkle P.0.,O.5 miles south of,

i nter se et ion Jalce s Prairie and T:~i b1Jy ro ad s, (T rop.d e a st) ( Si:':Sll board 10 rlli1es to C1..10 a), east sid.e Jake s Prail'ie 1'oad ,30 foet :,lOr Jcl1

of 1~ 03Jl in terscctiol1,8 ''i''lalnut t:re e Ii;.;. 'base; C01)lJo1' nail and y,~:;.sher.-------------------- ·. · .... -- 921.545 -0.264 921.281

83 ColI ins SellOoI, 800 f eot ~1ort:l of, Hink1 e store 8.11(.: P.O., 40 f eet southeast of,40 feet west of cen t e r of ~oad ,in lin e with t angent of road to northeast; iron post P. B. U. sta.:-::ped fl1039!' --------- ~ ,~-- - ... - - 1038.938 -0.258 1038.680

84 Collins Sch oo1,800 feet n orth of, Hinl:1~ Sto l'e a.l1d p. O. , southeast corner of porch, top of floor; copJ?er nail Gnc. washer. ----- .. -- 1040.617 -0.258 1040.359

85 Hinkle PO.,O.S miles north of, 0~1 Jak es Prairie ' road, sou th,!;7e st corner sec.13,R.6W;T.39 N.,350 f eet north of,on w~st side of road, opposite private T road east to vrhi te house ,16" black jack tree, in base of;copper nail and vlasher. ----- .. --------- 984.9 64 -0.254 984.710

86 Hinkle P.0.,1.2 miles north of, on Jakes Prairie r oad, a t pr i vate T road east,southeast corner of road,lOIl post oa.~ tree,in iJase of; copper nail e,.nd Ylasher. -------- 1057.863 -0.249 1057.614

87 Hinkle P.O.,1.75 miles north of, Oak Grove School,2.0 miles south of J old de serted farm houseol1 vre st side of road,125 feet southeast of, 011 top of hill, e a st side o'f road, 10" black jack tree,in root of; copper nail and vras her.--------- 1086.932 -0.244 1086.688

. Adjostment of Permanent Bench Mark S on Jakes Prairie Road at the Phelps-Gasconade Connty Line. The adjustment is to be made between the points 0 and S with the line tied back at O.

This section,O to s, is a spur line to the level net; it was run as a duplicate li~e of levels, that is:the line was run over the same rout~ twice~ but in opposite directions:i.e. o to S and from S back to o. ~ee page of this thesis for a discussion on duplioate lines of levels.

A N A L Y TIC A 0 JUS T N E N T o F POI N T S.

EleY.O - 969.7580 ,baerv.'n olrolilt a vIa 0 and N to M.

• o. 969.5;~1 Adjusted. _._------ 0.2229 Cor.

EleY.O- 969.74;0 Ob.erv. on .,ur return.

• 0- 969.53;1 AdJuet.

jF~:F~-1i, v s 0

- 0.2079 Cor.

<- - -O.S --> NoW for the adjustment of S.

Let X • the correotlon de.lfed.

The n:

and X & 0.0150 + 0.2079 ---2--

and:

x • -(0.0075 + 0.2079)

and fInally: X. - 0.2154 ft.the correctfon to be applIed to the obseryed eleYatlon of s.

Then: Elev l S = 942.5160 Observed. n _ O~ 2154 Correction. If s -= 942:3006 Ad j usted.

0\ ('II

IN ('\I

• C>

A {}

• • 0 -0 ... . ::110

o 8 S-ERV-EO-

0 oJ U S T

f t 0 w E R

E

... fC'I .... '" C>

D

S

fC'I

'" . o

L I NE.

l I M E

0\ .... C>

'" . C>

. 01

o fC'I

• CI>

DtAGRAM FeR THE GRAPHtC ADJUSTleN' OF POtHT S.

-106-

. 88

89

90

O.

s.

o.

(lak Grove 8cll0 01,1.5 mil e s south of ,on Jakes Prairie road,dim T road west, 200 feet north of,farm house on east side of road,200 ft west of,i~ fence corner, east side of road,14" red oak tree,in base of; copper nail and wash.er. ------r-Oak Grove S'cho 01,0.5 mil e s south of,on 3akes Prairie road,W. E. Wright's house,200 feet north of, at T road east and private road west thru gate, top of hi1l,center of roads; road elevation. ------ ..

Oak Grove School,on Jakes Prairie road,Dawson Township,Phelps-Gasconade· County Line,O.75 miles S. of,in front of School Rouse,125 feet west of, east side of road, 14" scaley bark hickory tree,in root of;copper nail and washer.--

Oak Grove School, in Dawson Twp., 0.5 miles north of,T road east, 50 feet south of,T road west to Red Bird road,20 feet north of, northvY'est corner of roads,lO" post oak tree,in base of;copper nail and washer. ----------------

Oak Grove School t O.75 miles north of,T road east on County Line, Jakes Prairie P.O.,2.0 miles west of,at private T road west on County Line, . sewell t shouse, 60 ft southwest of, Phe:bp·s-Gascol1.ade County Line,lO feet north of,north

99~. 372 -0.238

967.072 -0.230

900 .. 143 -0.226

969.758 -0.223

east corner of roads;iron post P. B. M. stamped "942 ft • ------------------- 942.516 -0.215

Oak Grove School,0.5 miles north of,T road east,50 feet south of, T.road west to Red Bird road,20 ft north of,northwest corner ·of roads, 10" post oak tree,Ul base of; copper nail and washer. ---------

-10'-

969.743 -0.208

999.134

966 .. 842

969.535

9420301

969.535

Adjustment of intermediate temporary bench marks between point N on the Red Bird Road 1.9 miles north of Royal P.O.,and point o on the Jakes Prairie Road 0.25 miles soutb of tbe PbelpsGasconade County Line.

The adjustment of thes~ intermediate temporary bench can be effected by scaling the adjustment corrections from the large diagram below and subtracting each correction from the observed elevation as it appears in the level field note book;or,if desired, the adjustment corrections can be oomputed by the following formula for this section,N to O.

o b 8 • .. '=-'=-=~.:::.=- jOb 8 ~ E I • v. N- 8 7' • 09:5 0 ~J tv E t • " • 0.969. 7 ~ 8 0

~ N Adj. 1\.l d ~ Adj.

E I. v. N· ~:~:~::: JL:-------::;-,--:;<f------:-lb E Ie v. o. :~::::~~ Cor. "'- 0.202:5 +-----2. -----~ Cor. __ 0.2229

OERIYATlON OF ADJUSTMENT FORMULA.

Correction c. for any intermediate point at distance d. from N and between the points Nand O.

N • .0 -0 -. 20

C -0.2023 d --0:0206-- = -2:4-

C = QLQgQ~_~ + 0.2023 2.4

C = -(0.008583 d + 0.2023)

C = (-0.008583 d 0.2023).

A D " U S T E 0 L I N E . 91 92 93

<> 0 0 r- t(\ 0-

• • '. 0 .-I .-I

GRAPH FOR TEE ADJUSTMENT OF INTER-MEDIATE POINTS BETWEEN N ANB o.

-ltJS-

(}

<:>

• • '"

Red Bird ~uadrangle. (Latitude 38 -38 l5';10ngitude 91 30 1 -9145 1 )

(Dawson Township)

Line from point on Jakes Prairie Road 0.25 miles south of County Line,thence west two-one half miles to point on Red Bird Road 0.5 miles south of Phelps-Gasconade County Line and 1.25 miles north of Royal P.O.

o Oak Grove School in Dawson Twp, 0.5 miles north of,T road east, 50 feet south of,T road west to Red Bird road,20 feet north of, northwest corner of roads, 10 " post oak tree,in base of;copper nail and washer.--------------- 969.758 -0.223

93 Oak Grove Church, BOO feet north 'east of,J.B.Cope1and's house,

BOO feet west of,in bottom of branch to north, north sidS of road,24" sycamDre tree,in root of; copper nail and washer. ---- 834.lB2 -0.219

92 Oak Grove Church,0.6 miles west of,T road south on line betv~en secs.2 and 3 produced north, 30 feet west, concrete cUlvert across road,north end wall, top of; paL~ted square.---------------- 895.903 -0.214

91 Oak Grove Church,l.2 miles west of, Red Bird Road,0.55 miles east of, Spurgeon's house,300 feet south-west of,on north side of road,twin trunk walnut tree, in root of; copper nail and washer. ------- 863.954 -0.209

N - Royal P. 0.,1.25 miles north of, on Red Bird Road,Oak Grove Church,

-1.85 miles west of,Phelps-Gasconade County Line,0.5 miles south of,Red Bird Ford across Bourbeuse River,llOO feet south of,at T road east to Jakes Prairie road,50 feet south of T road,on east side of road,14" red oak tree,

969.535

833.963

895.689

863.745

in base of;copper nail and washer. 875.093 -0.202 874.891

-109-

CON C L U S ION •

In conclusion, a brief outline will be ~iven coverin~ the main classes of problems which have been considered and the methods of solution and their appropriate applications to the problem at nand. Direct Observations of a Single Quantity.

AdJustment. Take the mean or the weighted mean of the observed quantities. Indirect Observations.

AdJustment. Write the observation equations and from them the normal equations; the solution of the latter ~ives the unknown Quantities themselves or the corrections to their assumed approximate values. The number of the observations equations will be the same as that of the observations; the

, number of the normal equations will equal that of the unknown quantities, which must always be less than that of the observations. Conditioned Observations.

id,ustment. Write the condition , equations in their ~eneral form and then in their simple form involving the corrections. From them form the normal equations, the same in nu~ber as the conditions. The solution of the normal equatiGDs gives a set of factors, called correlates, one for aaeh ciondition equation, from which the desired corrections to the observed quantities are determined.

In summarizing 'what has been stated above,there are in ~eneral two methods of finding the most probable values of the unknown Quantities in cases involvin~ conditioned quantities.

In the first method the cond~t~onal equations are avoided ( or illlminated ) by impressing their significance on the observation equat~ons,which reduces the problem to the cases as considered in Parts (II) and (III).

In the second method the observat~on equations are illiminated by impressing their significance on the conditional equations,when the solution !!lay be effected by the method of eorrelatives as considered in Part (IV).

The first method is the most direct in elementary problems, say with not ,more than three residual llnknowns,hut the second method by usimg conditional equations instead of observation equations and . completing the solution by the method of correlatives in connection with the results obtained from the solution of .the normal equations, greatly reduces the work of computation in the case of complicated problems with four or more unknown quantities to be solved. L The best methods of solv~ngDormal eguations,as I hav1

found them, are those given in Part (IV) pages 55, 56, and 57. -110-

APPLICATIONS OF THE METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.

ADJUSTMENT O~ ELEVATIONS IN DAWSON

AND ST. JAMES TOWNSHIPS,

PHELPS COUNTY,

MI$SOURI.

-B I'B L lOG RAP H Y,.

-11-1-

BIB L lOG RAP H Y •

References on the Method of Least Squa~es. Manual of Spherical and Pract..icalAstronoay.,Chauvenet.

.J.F. Lippincott Co., Philadelphia, 1885. Approx.imate Deterllination of Pronatlle Error" Appendix NO.13,Report

for 1890,U. S.CoastandGeodet,ic Sorvey. Theory of Errors and Method of Least Squares, Johnson •

• John \tiley & Sons, New York, 1893. Practical Astronomy as Applied to Geodesy and Navigation,Doolittle,

,John Wiley I'r Sons, New York., 1893. Precise Surveying and Geodosy,MerriJlan.John Wiley & SOIlS,

New York, 1899. Adjustllent of OtJservatioDs,Wrigbt anti Hayford. D. Van Nost.rand & Co,

New York, 1904.- 1907. Text Fook 011 Geodesy aouLeast Squares, Crandall.

John W.iley & Sons., New York, 1907. flandtJ ucb der VerllessuhgsktWde, I. Metzler, Stuttgart, 1910.11y Jordan. Ausgleichuugsreclmung. By Belliert. Tullner.,Leipzig ,1907.

Method 01' Least Squares, Merrillall, Wiley, New York, 1913. Method ot Least Squares, Comstock. Ginn, Boston, 1895. Metb.<Ml of Least Squares, Fartlett,Foston, 191~.

Iftu).ory of Errors and Least Squares,Weld. Macllillan,New York,191fi. Theory of Errors and Method of Least Squares, Johnson.Wiley 1892. Combinations of Observations, Srunt, Callbrid~e Uuiversity Press,1917.

Af ,licatioll of Least Squares to the Adjust.ent of Triangulation, Special Pub lication No. 28, tl. S.C.& G.Survey,1911).

Practical Least Squares, O.M. Lelar14l, McGraw-Hill Sook Co., New York, 1921.

GeOdetic Surveying and the Adjustment of Observations (Method of Least Squares)" E. L.Ingram, McGraw-Rill Book Co." New York, 1911.

Adjustment of Trianguiation,E.M.Douglas,Scbool of Mines Quarterly .. No.4 Vol.14 Colullbia Col1ege,New York.

Topographic Instructions of the U. S.Geological Survey., Wa.shington, 1918 Topographic, Trigonolletr ic., and . Geogetic Surveying,lUlson.

John ltiley & Sons, New York, 191'2.

Of the above treatme ou the sub jec·t of Least Sq~ares, the . last five mentioned were consulted as well · as the book furt.her up iI. · the list ·. by Welti. Tbebookb,y :Leland .is reCO.lll.llelltied as an exceptionally good book on tile subje.ct..

-112-

APPLIOATIONS OF THE METHOD OF LEAST SQUARES TO THE !f1.JUSTMENT OF LEVEL CIRCUITS.

ADJUSTMENT OF ELEVATIONS IN DAWSON AND ST. JAMES' TOWNSHIPS,

PHELPS COUNTY, MrSSOmH.

I N D EX. Abridg~d M~thod A for Solutton of Normal Equations • • • • • 55

" "Solution A with Nottls on the Process ••••••• 55 Abridg~d Method B for Solution of Normal Equations.. • • • 57

" "Solution B with Notes on the Proc~ss ••• • • • • 57 Acknowledg~ment.s • • • • • • • • • .• • • • • • • • • • • •• x

- -Adjustment of Int~rmediat~ Junction Points • • • • ••••• 18

" " " pointaL and K • • • • • • • • • • • • • • • • 18 19

.79,20 .~. ." " " " " " " " "

- -" Point P • • • • • • • • • • • • • • • • • • • " points N " Point Q

" " R

" " S

and ° • • • • • • • • • • • • • • •

• • •• .. . . • • • • • • • • • • •

• • • • .. . • • • • • • • • • • • • 21 • • • • • • • • • •

• • • • • • • • • •

22 23

Adjustment of Spur Lines PQ, NR, and asa • • • • • • • • • • 21 Adjustment of Interm~diateBench Marks • • • • • • • • • .2~,63

" " " Benchmarks Between A and I ••••••••• 26,53 -

" " " " " " I and P ••••••••• 29,56

" " " " " If

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " It " " " " " " " " " " " " " " " " " " " " " " " " tt " " " " " " " " " .- " " " "

" " " " " " "

A and D -

I and J .B and H

-Hand J -J and N

M and N Nand R -P and M E and D

-D and G C and F Hand F

-F and G -

• ••••• 34,92,93,95 · . . . . . . . . . 71 • •••••• • • • • 73 • • • • • • • • • • 76 • . • . . • · · • · 71 • • • .. . . • • •

• • •

· ••••• • 80

• • • • • •• 82

· • • • • • • 84 • ~ • • • • • 87 · . . .. • • • • • • 88

• • • • • • • • • • 98 • • • • • ••• .... 101 • ••••••••• 103

G and 0 ••• • • • • • • .104 .106 .1:08

o and S • • • • • • . .. . 'N and 0 ,. .. . . • • .. . ..

-113-

I N D E X A

Adjustment by the Method of Correlates • • • • • • • • •• 58 Adjusted Elevations of Junction POints(Summary) •••• 47,54 Application of the TheorY(M~thod of Correlates). • • • •• 60

B Back Solution(r Solution) of Normal Equations16,47,52,53,56,57 B~nch Marks(See Adjustment of Benchmarks •••••••• 63,25 , ,

Base Lines for L~vel Nels ••••••••••• 1,6 ,38,49,62 &:sat Methods for Solving Normal Equations •••••••• 110

C

Calculation and Tabulation of Adjusted Junction POints15,47,54 Carefullness of Obs~rvation. • • • • • • • • • • • • • • • ix Check Terms • • • • • • • • • • • • • • • • •.• • • • • • 57 'Check on Residual Values • • • • • • • . . • • • • • • • 15,17 Circumstances Aff~cting Observations . . . . • • • · . • • ix Closures in Arranged Circuits • • • • • • • • • • • • •• 39 Computation of Junction Elevations on Arranged Circuit. 9,39 Comparison and Application of the Two Methods ofL.S.. • 110 Condition Equations •••••••••••••••• §0,58,110 Correlates{Correlatives) and Corrections •••••••• 51, 56 Corrected Differences in Elevation. • • • • • • • • • •• 50

D Dawson Township ••• • • • • • • • • • • • • • 1, 6, 38,49, 62 D~rivation of Terms (Theory) for Level Net. • • • • • • •• 60 Description of Bench Marks and Their Adjusted El~vs26,29,31,34 Diagrams (S~~ Illustrations) •••••••••••••• vii,viii Differences,Observed • • • • • • • • • • • • • • • • •• 7,50 Dirrect Observations of a 'Single Quantity. • • • • • •• 110 Discrepancies Among Observations • • • • • • • • Discussion: Methods of Solving Normal Equations

• • .. . • • • • • • .. .

ix 56 59 Discussion": Methods for Formation of Normal Equations

Duplicat~,Parallel Duplicat~,'Lines of ~v~ls ••••• E

• 21,23

Elevations, Most Probable Difference of .. . - • • • • • • • 16

" " Most Probable • • • • • , • • • • • • • • • • • 16

" " Description of B.M.s and .AdJ.Elevs .26,29,31,34,54 , -114-

I K D E X

Elevations, Computation of • • • • • • • • • · . . . . . . " ", Difference in • • • • • • • • · . . . . . . .

Ellimination by addition and Subtraction • • • • • • • • •

39 50 12

Equations,Normal,Solution of. • • • • • • ,41,52,55,57,59,110 " " , Condi tional. 50 • . • • • • • • • • • • • • • • • •

" " ,Observation. .41 • • · • · · • • • • • · • • • • • •

" " ,Normal, Formation of 41,51 • • • • • • • · • • • • · Errors, Types and Occurrance of • • • • • • • • • • • Error, Probable error of a Single Measurement ••••

" ,probable Error of N Measurements • • • • • •• F

· . · . • •

ix 22 22

Figures (See Illustrations • • • • • • • • • • • • • • • • vii I

Formation of Normal Eqs.from Observation Eqs ••••• 10,41,42 ~ormation of Normal Eqs. from Condition Eqs. • • • • • •• 51 Forrr.ation of Normal Equations, Theoretical. • • • • • • •• 59

G

General Use of Theory among· Engineers. · . • • • • • • • • ix General Principle of Least Squares • • • • • Graphs (See Illustrations) • • • • • • • • •

I

• •

· . Illustrations • • • • • • • · . • • • · . . . .

· . . . . 4

· . . . • vii

· . . • • vii Indirect Observations • • • • • • • • • • • • • • • • 110

5 Intermediate Junction points. · . . . • • • • • • • • • • Intermediate Point~ on Level Circuits • • • • • • • • •• 4 Interrr.ediate Bench Marks (See Adjustmett of Benchmarks) •• Index

J

Junction points and Intermediate Points. • · · · • · • 5,15,18 Junction Points, Summary of • • • · · • • • • • • • • .24,47,54

L

Least ~quares,Principle of · • • .. • • • • · • • · • • x, 4,60 " " " ., Results of • • • • • • • • • • • • • • • •• x " " ", Use Among Engineers • • • • • • • • • • •• . x

Levels,Multiple Lines •• • • • • • • • • • • •• • • •• '2 " , Weights f or the Links • •• • • • • • • • • • •• 4

Level Bas~Lines • ••• • • • • • • • • • • • • • • • •• 7

-115-

I N D E X

Levels, Parallel Duplicate Lines of • · • • • • · . • • • • 21 Duplicate Lines of .. • . • .. • • • • • • . • • • • 23 " ,

Links • . . . . . . . . • • • • • • • · . · . . · . • • • • 50

Main Classes of Problems Mean,Weighted Arithmetic Measurements,Unweighted • Methods of solution • • • Most Probable Elevations Multiple Lines of Levels

M

• • • • • • • • • • • • • • • · . . . . . . . . . . . . . . · . . . . . . . . . . . . . . • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • · . . . . . . . . . . . . . . N

• .110 · . 4

• • 22 • .110 • • 16 • •

Nets,Diagrams of Dawson Township Net(See Illustrational) vii Normal Equations from Condition Equations ••••••••• 51

" "

" "Solution of1416,17,41,45,47,52,54,55,57.59,110 " "from Observation Equations ••••• 10,41, 42

o Observed Differences of Elevations by Links • e.. • • • • • 7 Observation Equations • • • • • • • • • • • • • • • • 8,41,110 On the Uniformity of Least Square Results ••••••••• ix On the Solution of Problems by Least Squares . . . . . . . 7 On the formation of Conditional Equations

p • • . . . . . • • 51

Parallel Duplicate Lines of Levels. • • • • • • • • • • • • 22 Plates (See Illustrations) ••••••••••••••••• vii Precision, Subsequent Measurements Dependent on Results Principle of Least Squares. • • • • • • • • Probable Error of Weight~d . .Ari thmetic Mean.

• • .. . • • • • • •

.. . • •

• •

x x

4 Probable E~eor of a Single Measurement. • • • ••• • • • •• '22 Probable Error of N Measurements. • • • • • • • • • • • • • 22 Probable Values of Unknown Quantities ••••••••••• 110 Problems, Solution of by Method of Least Squares • •• • •• 7 (P)Solution of Normal Equat~ons ••• •••••• • 16,45,52,56

Q Quantities,Summary of Residual Values ••••••••••• 15

" ", Check on Residuals • • •• • • • • • • • • • • • 17 .-116-

I N D E X Quantiti~s, Summary of junction El~vations •••• Quality of Obs~rvations,Car~fulln~ss in Obs~rving (q) solution of Normal Equations ••••.•••••

• •• 24,47

R R~marks on th~ Introduction .• ••• R~siduals (v), Ch~ck on ••••••

. .

. .

• • • • ix 17,46,52,56

• • • • • • • • • • ix . . . . . . . . • • 15

R~sid ual val u~s • • • • • • • • • • • • • • • • • • • •• 15 Rul~: Formation of Normal Equations from Obs~rvation Egs .10,41 Rul~-: Format1on of Normal Equations from tCondi tion Eqs ... 51 (r) or Back Solution of Normal Equations •••• 17,47,5'2,5~,57

S Singl~ M~asur~m~nt,probabl~ Error of • • • • • • • • • •• 22 Solution of Normal Equations •••• 12,16,41,46,52,56,110,Finas Stat~m~nt and Solution for Adjustm~nt Corr~ction • • • •• 3 Stat~m~nt of Conditions • • • • • • • • • • • • • • • • • 8 Subs~qu~nt M~asur~m~nts D~p~nd~nt on R~sults,pr~cision •• x Subtraction. Elimination by Addition,~tc. • • • • • • • •• 12 Summary of R~sidial Valu~s • • • • • • • • • • • • • • •• 15 Sum of W~ight~dSquar~s a Minimum. • • • • • • • • • • •• 58 Summation(l:)of W~ight~dSquar~s and th~ 'Conditions • • •• 58 Symbols.Nom~nclatur~ of 'Symbols us~d in Th~ory • • • • •• 59

T Tabulation:. • • • • • • • • • • • • • • • • • • • • .• •• 50 Tabl~ of Corr~18t~s and Corr~ctions to Links • • • • • •• 51 Th~ory Stat~d for W~ight~d Obs~rvations. • • • • • • • •• x Th~ory-: Application of the Calculus. • • • • • • • • • 58,59 Two M~thods of Findin~ Most Probabl~ Values of Unknown Quant110 Typ~s and Occuranc~s of Errors • • • • • • • • • • • ••• ix

U Uniformity ofL~astSqu8r~ R~sults • • • • • • • • • • •• x Unw~ight~d M~asur~m~nts. • • • • • • • • • • • • • • • •• 22 Unknown Quantiti~s,Two M~thod8 of Finding ••••••••• 110 U8~ of the Theory of L~ast Squ8r~sAmong Engin~ers • • •• x Us~ of Check T~rm8 ••••••••••••••••••• ~~,57

-117-

.~ N D E X V

Values, Check on the Residuals • • • • • • • • • • • • Values, Summary of Residuals ••••••••••••• v,Residuals ••••••••••••••••••••••

W

· . • 15 • • • 15 • • • 15

Weighted Observations, Theory of ••••••••••••• x,58 Weights for Level Lines, 1/ d. • • • • • • • • • • • • • • • 4,50 Weighted Arithmetic Mean, probable Error of • • .• • • • • •.• 4

... 118-

( 25611 )

:TIC COMPUTATION SQUARE ADJUSTMENT Rules for sides

stand outside anq!e: sid8S bQth reqd tow(lrcis WzqJe's vertex: Ilff'f-hand sid(J read-, r/ql7t hand sid! next cmd read T. Consider DicJr,tffen - 6/8 f 1/9

:U' JDick

(e) Sine Eauation

htJrica} exaMS goff"" Tram 'tJlimiM!y computatiOl1.

. second Groupe h.aV~~i 'J'1S cho!ltliJdin {I)e~ whMtht! diff.~or r is min~. h4n th" sign fW11t7JnS 3(71118.

Nohl thot th(J dlff. fbr rf'or a'?9/&lI o~(Jr 80' is minu:t.

1

1 J

Sides .... h fOb

r -&/ot'ID -~/J'r·11

-1 -abr'b -:om r"/o -Ok r 11

Anq/8 Sine I)/ff.fbrl.' 8:J"-.<H3 - ~:16 9.9S.7Jf6JftJ 1'022 ~-18J7 3.lJ130J:JO r~.6 "t5'i2-~7.0"" S,85107.3f na9

l' 8.IJfjJ/572B

.. 0"-09' 14. '16 ~"''t 1'2+.8 't5'-,3B'-.:3+. :9<: 9.8!5.y.()fS76 "'2lJ.e 815 . 23 -07. '2 9.8980787 -2.

t - S,(!JS15906 t r 9.661:572B

Error-ISO

I £:s::t/:;; 1.33

r2.12 1'278

-a.S) 2.97

1O.7T

Fronf and back angles Front ang/M T Back (l'7!l/es-

I~l" 'SIne - -$ - 9.997"fo6<f.Z - t~ - 9.IJ130#02 - ~15-:::; ~O7'7 9,RAil5B21

- -23 - 9.B09Lr520 - -6f ~ 8.8IYrasT5 - - .:o~07=

s.ee/lJ821

fD ll. { 0--""'-=->.="'-....... 2 ... -.209"="'~ } { .... .,...... -249'1>f.206'1o-206"b-.a20"IJ f:0206b

divided fy JOO, S use o'?)':3 of'4 angle equaTion

, problMl IMve oiJt ((I)

ians

ComlJirM like trJrms alg8btYrical!y; thM

0~/.BOt227·hf.022°b-.452"1ot2.y.B'Jo-2CJ931I1'.228""~"t2(){J'1.-:02UIl

of Condifkm Table I of'lI Cor;reJoj8S I ! J! 1 J J 70 t'/o -Ollt-t,.-IJ.'t0/r (from aJ iii (k) (JJ m) In)

__ A~/"u

o .,."", -DJzt~Jz "brO" (f'rom bJ 1 - f'lrff'l fIIr(,,) f' .... lh)if'lrW l+az7.'iI--n.DA'i -G!,/i9 t,'9.fln7 d 1

~o tllo ""'ld,,,-Ib.,.O,,, (f'romoJ '10 +/ f'/ 1ta2+61+1l275 -0.589 l+a740- "'0.¥~6 -I . L __ I"fo I-f It'll t-Q'052Hl275r-£l063 I 1-/.3/59 -1-1.897 -I

-1 Title Pagt' 6 Contt'nts 2 Introduction 4 Front Platt's in Foldt'r

118 Body of Tht'sis 1 Rear " Plate

132 Pages Total.

Thesis C. E. S. Bardslt'y., 'Candidate for the Degree

of iCi;vi l En~ineer.

GEODETIC COMPUTATION LEAST SQUARE ADJUSTMENT

T'!)'lor2.:--______ .lBrowning

Rules Tor sides ofr ~

D~3~ ____________ ~OE~

Male Eduations

L -St"/=No. of'anq/e equations L=No. of'lines /nrYqurYJ =6. _ S=No. of'stations =l:j. Then (j-LffJ=3 aryle equaTions

L -25 f.J =No. of'sin8 equations l (J-(2X4Jt3=/sine equations J ~

StaTion Side Observed Ana/B Cor.f'orAnq/e CorrYJctedSaAna/e

leo' 00' ~.(J4 2.74 leo' 00' -0 .SO

Dick -"bt'/3 +0" 09'-/+.16 -0.90 +0' 09'-1328

179''"59' --s8.29 +I. +:'J ISO' -00'-00. 78

Stand oufside anq/t:' sid8S both rtlad fow(I('(is t a(lqlt:'3 vflriex, Ilff'f-hand sidfl rtlad-,r/qllt hand ~ sid! next and rtladt: Consider DicH,tfftln -a/s f '13

~U' .:JOic/r

~i~ (e) Sine EQuation

Sides

Fronf and back ang/fJ3 Front angJ~ 1-Back a'?!}/t:.S -

l£Jk

-3,,'1'-"

f---r----.-[

+- - /Of'lo ~5'-12-:37.0+ 5.8510731 f20S t2J5 - I He -0 !J.lXJI0777

This second 9rtJUPfi h.aye~i '/7$ 8din (/)e)(, when the ~or I" is min~, hen the sign

I"8/7I(7l17swm~

/VoftJ thol' the diff. fbr I"/'or a'?9'M o~er SO·isminU3.

1 J

""bf '/3 %f'/o

-Ok -t""

-t 8.&515728 9.tJaI6821

+O"-OS' 14. '15 9~"t."I

~ 38'-:3Lt::90 s.6!540575 96 . Z3 -07. 52 S139807tj7 t - Me/590B

9,8615726 Error 180

+2+.8 r20.8 -2.0 t

-0.91 - -23 - 9 .809+520 -2.97 - -81 9.8l5+0515

ITU71 ~ - - 9.S8B076t' !J.tJtJl!582J

r ;r,::;'''s / 1 ""- a~8 L ____ =============----==:+======v:::'" X::"::' :::::":::::'==------1 Con use a'7Y.:3 of'4 angltJ equations This probll!lm l(JUvtJ out (tf)

{o)Normol Equations k Im(J

m ua

y. Z I~m m ,,,,, I

I 2 3 1

~. Ab30Jute Torms

arr03 , umTTl in I

1 £guotions of Condifion

{f')o--3.'t5-·IO+''.2..- 0fJ.t-tc:'lat Ok (from al (9) 0- r"t.70"'lo", ': _,;:' N (t"rom bJ rh) 0- t2.7~"'lo t'lo ~" ""t ~. (;

Table of' 2 Co~, GI'-'.,' ov / 2 ..

(vI (k) (mJ (nJ .... ' _.m.,; yoluo I... '£J Ifaz75!-<t063 1-0$89t3007 I~

1-0.58.9 +0740- Iro.~26 1-006.3 ' -1-1.697

t6.000 -2.000 +2.000 fa8'<7. 3.'<50 ..... >n t L--__ --, 11Q.063 1tQ~ to -lrl.Z:

-2.000 t6.000 1'2.000 0.863 • t4.700 ..... --1'2.000 1'2.000 t6.o00 tOJ22 t2.740 ___ ---to.$'<7 -0.863 to. 122 TO.5/7792 -1.800 - -t

(PJ Solution of' Normal Equafions

I I I I 2 3 ~ ~ PI n;.ooo 1-2.000 11"2.000 ItO.9+7 -3.'<50 P:. (-0.165; 1ta333 1-0.333 1-0.1578 fO.575

~ 1+5.:333 1-t2.667 1-ll547~ '3.550 J'+ .f) JFi7.Z;: 'r0.500 ItOJ028 -0.886

P6 ItLt:OOO -0.0800 1'2//5 -P. "- '(-0250) 1-0.0200) -0.529

f7 'rTa310B() -O~'t r3.007

(5) Anq/e Corrections

:k: .U ... " 1M OS

~ , ",

..J. rom or cv-ror In

Fl"'Orn I or vretJr "

ItO.5B9 -C I u 11"025'# to./. -11'0.86+

t-o.589 -al . -1-1.217 Ita.338

0275 1-0275

1-0063 1-G58.9 1:066 -1-a58B

!t02Zi

1 (q) 'Add ".:7' -~

2. qJ 11'6.o0G +2l)OQ 1-0.863 '4." 00

+0.667 1+1I .31S~ . ISG If5.333 r2.6/ 73 't3.550

ao t/ It( . -t2.740 --0 QIo -0.8/ 1-( ~7 rl. /50

(r)

11 I Ll--!:f-~I"~o .. S~7~S-F.·~ 0.:,=,tl!=-'6-j~_ -'7': a;~:~;.o~.9o--+l..!. t"""".3 .. to=7----j L-==: '" a47S 1'0.308 ,- ,~,

n. ·alge .,0.294 1-0.589 r. 0021 ·a~ f'b . Q275

"_ I I

,or Anal" of FJk Anall! oflJrownina MoM of Dicit 1

~ -Cor "0· - /.271 _~D" --0254- Cor '/3 -"1'0 160"'- fjopypbovt! 2.0rrtlGfjrr::/n;!,hg:frr;e:r;Pjg~~/ons l _______________________________________________________________ -'

r+-"'Cor"'--"'IO!!..-"-:-1'';;a-ii47'2~6'--+'t-'Cor~..!·/!..J »'--:'-/.7"'27f,1.f7;---jIt!-'ICor=-..!D~:o_-_-~ u.,5~8'86H 'I ~~'s'1:t,n 6!/u~'i:;S. s ' J 0:7 J Jr' - 0.845 -/.4t71 0.Jt-26 I- l

3 OOA'

AnqllJ at Dick Anq/t) at fjrownino Ana/I! af Tov/or Cor~~ = -0746 -Ca- 3

/, s 1'1.217. k20r'k - tf!..275 rCor '/:I = - 0.160 -t Ca- '" = TO.96't tCor '% - -Q083

0.908 T2J8I t 0212 I-

DIRECTIONS FO/low mr;jorOP8r:oTlons dITJhab8fical!J and the

d6tal/" num6Jficaljy . Computation ty ;u... YEAGEJ{,.

YocatiOflfl/ SttdtJnt

MISSOURI SCHOOL OF MINBS AND METALLURGY

applications of method of least squares to the adjustment

Documents