calculations to determine at what point in the side of a hill its attraction will be the greatest,...

Calculations to Determine at What Point in the Side of a Hill Its Attraction Will Be theGreatest, &c. By Charles Hutton, LL.D. and F. R. S. In a Letter to Nevil Maskelyne, D. D. F. R.S. and Astronomer RoyalAuthor(s): Charles HuttonSource: Philosophical Transactions of the Royal Society of London, Vol. 70 (1780), pp. 1-14Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/106365 .

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P~~~~~~~~~~~4 d 1 LOS PH it:

T R A N S A C TI- O N S.

1. Calculations to determine at wbat Point in the Stde f a

Hi/I its Attraaion will be the greatetf &c.

BY Charles Hutt6n., fLL.JD. and F. R. S. In a Letter to Nevil Mafkelyne, D. D. F. R. S.

and Aftronomer Royal.

Read Nov. it, 1779.

3DEAR SIR1 Royal Military Acad.

A s the experiment of determining the univerfal attraffion of matter, which you lately condu~ted

with? fo much accuracy and ficcefs, is of fo great im-

VOL. LXX. B portance



afi Do&or HUTTON'S Determination of the portance and curiofity, that it may- probably be here- after repeated by the learned in other countries, and in other fituations; and as the utmoft precifion is defirable in fo nice an experiment; I have compofed the inclofed

paper to determine the beft part of a hill for making the obfervations, fo as to obtain the greateft quantity of

attraalion I have no doubt, Sir, that the determination of this

point will appear of fome confequence in the opinion of

one who has the improvement of ufeful knowledge fo

much at heart; and if the manner in which it is here made, meet your approbation, I iliall defire the favour

of your prefenting it to the Royal Society..

I have the honour to be, &c..

i. The



Place of great Attra6ion. i. The great fuccefs of the experiment, lately made

by the Royal Society, on the hill Schehallien, to determine the univerfal attradtion of matter, and the im- portant confequences that have refulted from it, may probably give occafion to other experiments of the fame kind to be made elfewhere: and as all poflible means of accuracy and facility are to be- defired in fo delicate and laborious an undertaking; it has occurred to me that it might not be unufeful to add, by way of fup- plement to my paper of calculations relative to the a- bove-mentioned experiment, an -inveftigation. of the height above the bottom of a hill, at which its horizon- tal attra6tion fhall be the greateft; fince that is the height at which commonly the obfervations ought to be made, and fince- this bell point of observation has never been any where determined that I know of, but has been va- rioufly fpoken of or gueffed at, it being fometimes accounted at 3 and fometimes at -1 of the height of the hill; whereas from thefe inveftigations it is found to be generally at about only W of the altitude from the bottom.

2. Let ABCEDA be part of a cuneus of matter, its fides or faces being the twoo fimilar right-angled trian- gleS ABC, ADE meeting in the point A, and forming the indefinitely fmall angle BAD. Then of any feafion

B z bced,



4 Doaor HUTTONS Determination of the

bced, perpendicular to the planes ABDY

c 'and ADE, the attraaion on a body at A in the dire6lion AB, is equal to the conflant quantity ss; where s = fin.

BAc and s =fin. <BADD to the ra- dius T.

For, firft, fince the magnitude of the flowving fec-

tion is every where as AbP, and the attra6tion of the

particles of matter inverfely as the fame, or as AP~ therefore their produ&t or I or I (a conftant quantity)

is as the force of attraation of bced. Then to find what that quantity is. Put AB = a, and

Rc = x; then BD or CE (the diftance between the two planes at the diftance AB) is = as. Now the force of a

particle in the line CE is as in the diredtion AC, and

therefore it is as AB in the dire&tion AB; consequently

the force of the whole lineola CE in the direation AR AB. CB is -.C--; and therefore the fluxion of the force of the

fedfion BCED or f is=ACE. BC _ ___ AC 3 ~ 2 1x

and the fluentgives f i sx BC ss for the AC

,attraiftion itfelf.

3. 'To



Place of greate/i Atlra7ion. 3. To find now the attraftion of the whole right-

angled cuneus on a body at A in the direftion AB.-

Since the force of each fe6tion is ss by the laft article, therefore the force of all the fe6tions, the number of

BC them being AB or a, is ass = s. -AB ' - the force of the

AC,

whole cuneUs A1XCEDA.

4. To find the attraftion of the re6t- angular part ABCD on A in the direfion AB ; ABCD being one fide of the cuneus, and AD its edge.-Put AD = BC = b

and AB = x. Then, the force of any feation as BC being

always as ss by Art. z, the fluxion of the force or f will be _ ssa = s X BC = Sx - - _ ; and the

AC 42 Vb2 + VZ

fluent isf= bs x hyp. log.of X+4zs S.- S C x hyp. log..

BC the attraffion of ABCD.

c To find the attradion of the

right-angled part BCD of a cuneus whofe edge paffes through A the

A.;1} P . place of the body attrafted.-Put.

AB a, BC ba, BD = C, DA-d = a-c, DC=e, AC -g,

and DP -x. Then, the force of any fe~tion Po being

fill as ss, the fluxion of the force of the part' DPQ isf

=SIx $- x Pi - = -- ; and the corre&6 fluent AQ~ Vdy 1 +2 cdxet xt

when:



6, DoFor HuTTON Determination of the .~~~~~~~~

hen X c is f x g-d -x hyp. log. ,

the force of a body at A in the direffion AB.

e 6. To find the 'ttraaion of the right-angled part BCD on the point

3Z~t z -A.-Ufing here again the fame 'ISi notation as in the laft article, we have

PQ_ sx C -S - =" S- r _ * The correa fluent f ~~AQ_ lcd 2c,'I dx + {

of which*, ashen Amx C, is bes fe+eg-d - ..g a +, .

. .

-..

,

f ee x-d+e -x hyp. log. Ae+e-de

7. To apply now thefe premifes to the finding of the place -where the attraftion of a hill is greateft, it wrill be neceffary to fuppofe the bill to have fome certain fi- gure. That pofition is moft convenient for obferving the attraftion, in which the hill is moft extended in the eaft and weft direftion. Suppofing then fuch a pofition of a hill, and that it is alfo of a uniform height and meridional feffion throughout; the point of obfervation muft evidently be equally diftant from the two ends.

BXut inftead of being only confiderably extended, I thall fuppofe the hill to be indefinitely extended to the eall and to the well of the point of obfervation, in order that the inveftigation may be mathematically true, and yet at the fame time fufficiently exa& for the before- faid limited extent alfo. It will alfo come neareft to the

I practical



Place of greatefi Attracfion. 7

pra~tical experiment, to fuppofe the hill to be a long triangular prifm, fo that all its meridional feaions may be fimilar triangles. Let therefore the triangle ABC re-

B prefent its fefion by a verti-

.. ,. .

Cal plane paffitig through the

meridian, or one fide of an indefinitely thin cuneuswvhofe

|. .. i edge is in PG; or rather A. G C 6PBCoP the fde of one cuneus,

and PAG the fide of another their common edge being the line PG perpendicular to the bafe AC; P being the required point in the- fide AB where the attrac~- tion. of the feaion ABC, or indefinitely thin cuneus, fhall be greatef in a dire6tion parallel to the horizon AC. And then from the foregoing fuppofitions, it is evident that in whatever point of AB the attra'tion of ABC is greateft, there alfo will the attraetion of the whole hill be the greateft.

8. Now draw HPDEF parallel to AC; and AH, PG, BI,

cu, perpendicular to the fame. Then it is evident that at the point P, in the dire6tion PF, the attraction of

PBCGP is affirmative, and that of PAG negative. But

PBCGP is- PBD + BDE + PFCG-EFC; and PAG = PHAG-

PHA. Therefore the attra6tions of PBD) BDE, PFCGI

BIa, are affirmative; and thofe of EFC, PHAG, negative. Put



8 Donor HUTTON'S Determination of the

Put now Bl =a, AI-b, IC-C, AB = d, BCe, AC=g=

b + c, and PG = X, the altitude of the point P above the l)ottom. Alfo let s = the fine of the indefinitely fmall

angle of the cuneus to rad. ;; and q2 _Va~g1 -abgx+daxi. Then by Art. 3, the attraction

* BD= fPBD iS J PD s BP x J

BPA t is s

PuPG Jb x

By Art. 4, the attrafcion

PFCG is S .-PG x h. -1. P-PC = sx x h. 1 g Ax+| of ~~~~~~~PG a

PH +PAI PGAAH is S . PG x h. 1. = sx x h. 1.-.? J

By Art. 5, the attraation of EFC is

EF. FC PEI. EF EC2 * EC.PC + PE.EF Ec x PC -PE - x h.l. _L+_

E' EC PE. .EC + PE. EF

sc a- cg_ - eqq + aax -bcx - * x qq,-g, * a--Ng X - gc . h1. 1. -- . * re g. c+e.a-x

Lafltly by Art. 6, the attraftion of BDE iS

BD. DE PE . DE BE2 + BE. BrP-PE. D B . B z - + x hB- PE. BE -PE. DE

SC - .v ec + dec-cg -..a - x d-g + g- x hl.- - re e eg -cg Thefe quantities being colle6ted together with their

proper figns, and contra6ted, we have fab C + , x yp. log. l +d|

et. . +_ .t. . ̂ .

S 3 ___x h.1. :Y

+ J} g. ee-c. a-x for the whole attraction in the dire6tion PE.

9. Having



Place of greatei A/ttra7ion4

9. H4aving now obtained. a general formula for the meafure of the attraction in any fort of triangle, if the particular values of the letters be fubftituted which any praffical cafe may require,- and the fluxion of- this attraafion be put =o, the root of the refulting equation will be the required height from the bottom of the hill.

lO. But for a more particular folution in fimpler terms, let us fuppofe the triangle ABC to be ifofceles, in which cafe we {hall have d=-e, and g = 2b = zc, and then the above general formula will become

[24 a? d +X x h. 1. -a+qq-_ X

,S x 4 dd bI+ 9 is agsX. Xb3 x h. d .

d3

for the value of the attraffion in the- cafe of the ifof-

celes triangle, where q1 is V/4 a2b- ab2x +dzx

And the fluxion of this expreffion being equated to o, the equation will give the relation between a and x for any values of b and d, by a procefs not very trouble- fome.

i . Now it is probable that the relation between a and x, when the attraction is greateft, will vary with the various relations between b and d, or between b and a. Let us therefore find the limits of that relation, between which it may always be taken, by ufing two particular extreme cafes, the one in which the hill is

VOL. LXX. C very



To Docor HUTTON'S Determinationi of the very fleep, and the other in -which it is very flat, or a very fmall in refpea of 1b or d.

i 2. And firft let us fuppofe the triangular fe&lion to be equilateral; in which cafe the angle of elevation is 600, which being a degree of fieepnefs that can fcarce- ly ever happen, this may be accounted the firft extreme cafe. Here then we ihall have d= 2b =3aV3, and the

2a- r- x formula in Art. i o, will become s X : + X i

h.Lo a r-x a+, x h. +2r+x for the value of 3x 4 a- x

the attraftion in the cafe of the equilateral triangle, in

which r is = V/a - ax + xs.

13. Or if we take x = na, where n expreffes what part of a is denoted by x, the laft formula will become

sa x I "-n ZVi-n + n' + n x- h. L. 2

+

4 x h.. 1I+n+2 + for the cafe of the equila-

teral triangle.. 14. To find the maximum of the expreffion in the

laft article, put its fluxion o, and there will refult I + 2-X + I I-n + XX

this equation i + h =h.l.

+ + 24/I-fl +f;Zthe rootof which is n =*5 I99o.

Which flhews that, in the equilateral triangle, the height from the bottom to the point of greaten attraction, is

only



Place of greatej Attra$ ion. tt

only ?' th part more than ? of the whole altitude of the triangle. And this. is the limit for the fteepeft kind of hills.

x 5. Let us find now the particular values of the meafure of attraffion arifing by taking certain values of n varying by fome fmall difference, in order to difcover what part of the greateft attraftion is wanting by obferving at different altitudes#

I 6. And firft ufing the value of n (25 1999) as found in the I 4th article, the general formula in Art. 13, gives sa x I0763700 for the meafure of the greatef at- tra6tion.

i7. If n =I0 or x k -,.a; the fame formula gives

20X17 -/79 + 6 h. 1. 17+ 2V79 + 7 h. 1. 1 .+L2LV9 _ - X 9 + 6hO 10~~~ 1

sa x I 0 7 0 2 5 1 2 for the attraftion at -I- of the altitude, which is Something lefs than the other.

giSsa IS. If n i=2; the formula gives 2 X I6

i/76+ 8h. 1. + 4? +3h. 1.7 + 7 sax. 22423'Z

for the attraation at -L or 5 of the altitude; lefs again than the laft was.

I9. If n = -; the formula gives ?sa x : 3

/3 2h.l-3+ 3 h+ l3+2V/3=sax *9340963 for the attraftion at half way up the hill; ftill lefs again than the laft.

C 2 !2 o. If



12 DoIor HUTTON'S Determination of the 6 so. -If n =-0 = 3; the formula gives 520 x :14- 5 ~~~~~~~20

V76 + 12h. 1. 97 +2h.1. 27 =sax,81o9843

for the attraffion at -6- or I of the altitude from the 10 5

bottom; being fill lefs than the laft was. And thus the quantity of attra6tion is continually lefs-'and lefs the higher we afcend up the hill above the 25 i999 part, or in round numbers z252 part of the altitude. Let us now defcend, by trying the numbers below *252; and firft,

2i. If n = 5= ; the fame formula in Art. 13,

gives j-sax :7-VI'3 +h.1. 7?+2-. + 3h.1 5+1V, 3 - 8 3 . 3 sa x I0763589 for the attraafion at 4 of the altitude; 4

and is very little lefs than the maximum. 22. If n=A -= -; the formula gives Isa X 9-

V2 i + 2h. 12 h.l. 3+4/2! - =I9-V2 I:

+ 2h. 1. 23 + 5V/2 - sa x io6846z2 for the attraaicn 2

at -L or j. of the altitude; and is Something lefs than at 4 of the altitude.

23. If n = o; the formula gives x : X 9-V91 +

2h. 1. -9 + 29 + 29 h. . - 9-sa x .9986I88 fot 3 2

the attraction at I of the altitude; fill lefs than the laft was. And, laily,

24. If n = o, or the point be at the bottom of the

hill;



Place of greateJ! AttraSion. T3

hill; the formula gives sa X 2 + h. 1. 3 =sa x 774653t for the attraffion at the bottom of the hill; which is between a- and I of the greateft attraftion, being fome- thing greater than j. but lefs than I of it. X ~~~~4

25. The annexedl table exhibits a fummary of the 6 ] calculations made in the preceding

<5- 934o963 2 articles; where the firft column: 14o I0224232 f fhlews at what part of the altitude

25 I37025 o 'a of the hill the obfervation is made;

T"75 I0763700 0

4 I0763 589 {7TTCthe fecond column contains the 2 IO684622 I correfponding numbers which are IU 99S6I88 'X proportional to the attraation; and 0 774653I T the third column fIhews what part

of the greateft attraction is loft at each refpeftive place of obfervation, or how much each is lefs than the greateft.

a6. Having now fo fully illuftrated the cafe of the firft extreme, or limit, let us fearch what is the limit for the other extreme, that is, when the hill is very low or flat. In this cafe b is nearly equal to d, and they are both very great in refpea of a; confequently the formula for the attradtion in Art. i o, will become barely

s x x x h. 1 2-X + 2 a-x x h. 1 2a-x;

the fluxion x aw

of which being put = o, we obtain o = h. 1. 2-x

2 a -x 2a-X -Ia- zh. l -- h. 1. h

. .-- he 1.-L a-x x a-x X.2a-x

hence therefore a- 1 _ x. 2a -X, and x = ax i -V/-

2 9 9 a.



14 DocIor HUTTON'S Determination, &'c. .929a. Which ihews that the other limit is -e9 ; that

is, when the hill is extremely low, the point of great-

eli attraiion is at -L9L of the altitude, like as it is at -245L when the hill is very (teep. And between thefe limits it is always found, it .bcing nearer to the one or the other of them, as the hill is flatter or fleeper.

27. Thus then we find that at ! of the altitude, or very little more, is the befl place for obfervation, to have the greateft attrition from a hill in the form of a triangular prifim of an indefinite length. But when its length is limited, the point of greateft attraction will defcend a little lower; and the Ihlorter the hill is, the lower will that point defcend. For the fame reafon, all

pyramidal hills have their place of greateft attra6tion a little below that above determined. But if the hill have a confiderable fpace flat at the top, after the manner of

a fruflum, then the faid point will be a little higher than as above found. Commonly, however, - of the

altitude may be ufed for the belt place of obfervation, as

the point of greateft attra6tion will feldom differ fenfibly

from that place. And when uncommon circumftances may produce a difference too great to be intirely neglea-

ed, the obferver muff exercife his judgment in guefling at the necel'ary change he ought to make 'in the place of obfervation, fo as to obtain the befl efife which the

concomitant circumftances will admit of. ..l .- is



calculations to determine at what point in the side of a hill its attraction will be the greatest,...

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