bartels johnston

8/10/2019 Bartels Johnston

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GEOMAGNETIC TIDES IN HORIZONTAL INTENSITY AT

HUANCAYO

BY J. BARTELSAND H. F. jOHNSTON

PART I

A bstract---Follow{n an introductory survey of the main features of the solar and

lunardaily variations, and , in horizontalntensity,H, in January t Huancayo,

da.ysith onspicu,us.unarn,,fluences,eomagneticides,re /scusse/t.separation

of $ and [. on such big-t.-days is_attempted. Daily ranges 4 in H are then introduced

., ' ' r c . ' " ' ' '

corpuscularadiationP.

Variousmethods or studying _are compared. Lunar semimonthlywaves n the

rangesA are computed nd discussedn their changewith season nd sunspot-cycle.

In the months November to March, when k is larger than in the rest of the year, [.

and$ increase,n theireffects nA, proportionallyo each ther romsunspo.-m{nimum

to sunspot-_maximum,ut aroundJune, when [. s small, t doesnot_partmipaten the

change f $ with_he s.unspot-cyc e.he day-to-dayvariabilkyof S and [2 s studied

in some detail; S and [. fluctuate rather indepen4ently of each other, and the relative

fluctuations f [. seem o be greater han thoseof S. The eliminationof the lunar effect

Az {s described; A -A r..) -As is proposedas a measure or W.

A more extensivesummary, ncludingsome esults o be described n Part I , has

beenpublished lsewheresee of "References" t end of paper].

. Introduction

The time-variations in the horizontal intensity H recorded, since

March 1922, at the Huancayo (Peru) Magnetic Observatory of the

Departmentof Terrestrial Magnetism, Carnegie nstitution of Wash-

ington, are known to show singularly arge amplitudesA in the solar

daily variations$. In connection ith a plan [21] o use$ for a day-by-

day measureof the intensity W of solar wave-radiation, t became

necessaryo eliminate the effectsof the lunar daily variations [_. In

contrast o the experience t other observatories here I_ s small and

canonly be extracted rom many hourly values, . in//at Huancayowas

found to be of exceptionalmagnitude,not only in absoluteunits, but

even elative o $, thus offeringuniquematerial or the study of [_, or

which the name geomagneticidesseems itting.

For conciseness,bbreviationsand symbolshave been usedas listed.

in 2. The main resultshave beendescribed lsewhere4] in less ech-

nical language; he background or this study is given in two recent

books 1, 2]. Generalstatisticalmethods or the determination f $

and [_have recentlybeenauthoritativelydescribed y S. Chapmanand

J. C. P. Miller [3]; the methods sed n the presentpaperdiffer n so ar

as hegreatermagnitude f [. in/-/at Huancayo ermits ne o gobeyond

the harmonic nalysis f daily variations, nd to determine he influence

of [. in the semimonthlywaves or every hour of the solarday. As will

be described n Part II, the latter method (developed from what is

described s "van der Stok's" method in [ ], as recently applied by

M. Bosso asco8], J. Egedal 19],and W. J. Rooney 20]) provides

direct approacho the study of the changeof [_with day and night.

269


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GEOMAGNETIC TIDES, HUANCAYO 71

meanperigee nd apogee s tabulated n the "Mondtafeln" 6],

and the quartersREC and NEA are the sixor sevendaysbetween,

on the average,6.777 days.

(6) M2,N2,L2,2N=partial unar ides, eePart .

(7) Seasonal roups: December-solstice--Novembero February;

equinoxes--March, pril, September,October;June-solstice--

May to August. In order o reduce ccidentalrrors, moothed

monthlygroups aveoccasionallyeen ormed n the sensehat

"January"means (December+2January+February)/4],tc.

(8) H= horizontalorce t Huancayo,xpressedn theunit -- 10 cgs;

occasionallyhe unit 0.1= 10 6 cgs--microgaussas foundmore

convenient.

(9) $ and .,solar nd unar ailyvariations' neach ay,

a superpositionf a quiet daily variation$ and a disturbance

daily variation$D. On undisturbedays--see 10)--$D n H is

negligible,o that $$. Average ariationsor a numberof

days,as distinguishedrom thoseon single ays,will be denoted

by $* and [.* in 6; [.* s definedn 7. Dayswith conspicuous

lunar influencewill be called "big-k-days" 4, 5).

(10)C--International agneticharacter-figure,etween.0 and2.0,

used as a measure for the intensity of the solar corpuscular

radiationP (particles) eaching he Earth: Undisturbed ays,

definedhere by C


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272 J. BARTEL.? AND H. F. JOHNSTON [vm.. 4s, No. al

Txm 1-B--AverageZiirich relative unspot-numbers,

Dec.- iEq ui- June

Group Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. solst. nox solst. Y

Min 6 10 6 8 10 10 7 4 6 7 0 10 9 7 8

Min 13 27 32 22 28 28 25 22 28 31 22 19 20 28 26 2.

Maxa 72 70 56 51 60 63 49 60 58 64 60 66 67 57 58 6

Max4 91 93 78 99 99 98 115 102 94 92 90 98 93 91 104 9,

Min 9 17 18 14 18 18 15 12 16 18 14 13 13 16 16 1,

Max 83 83 68 78 82 83 86 83 78 80 77 83 82 76 84 8

All 48 52 43 46 50 50 51 48 47 49 47 51 49.5 46.1 49.7 4

....

7,9

.4

80.4

484

(0 to 5)h; A is the lunar effect on A, and `4-AL=As. The

deviationsof .4 and As from their averages or the calendarmonth

are AA and /XAs. Rangesof instantaneous alues, published

as RH in the "Caractre magntiquenumtriquedes ours" for the

years 1930 to 1939, will be comparedwith A in 13.

(13) Lunar semimonthlywaves n the rangesA: aa cos2v+b.4 sin 2=

c sin (2+), see 14.

(14) Probable error-circle adii 0 of cloudsof points in harmonic dial:

See 15.

3. A preliminary escriptionf themain eatures f L

The details of the calculations will be more easily explained by be-

ginningwLthan advance ummaryof the main featuresof $ and [. in

H at Huancayo; heseare combinedn Figure 1 for the monthof January,

when L is great. The upper half of the diagram refers to sunspot-

minimum, the lower half to sunspot-maximum Min and Max, see

2(11); averagesunspot-numbersor January R=9 and R=83]; the

solaractivity is pictured,at the left, by two spectroheliogramsaken in

calcium ight at Mount WilsonObservatory. The phases f the Moon

are picturedby schematic isks or --0, 1, 2, . . . 24 hours.

The two frieze-like diagrams at the upper and lower borders of

Figure1 are derived romall observationsn undisturbedays. They

show he systematic uperpositionf $ (secondow) and 1_ third row)

to form the observeddaily variation ($+1_) (first row). The ordinate

scale s indicatedby the vertical lines at the right which give 296vm

oneper centof the average/-/at Huancayo. Only the average ariations

are shown in the friezes, as if $ and [. were completelydetermined by

the season here, he monthof January), he sunspot-number,nd, in

the caseof L, by theage vof the Moon; he friezes retherefore chematic

in so far as they showneither he additional rregular luctuationsrom

day to day (4) nor the change f L with the distance f the Moon from

the Earth (Part II).

$, _,and ($+ _) are picturedas departuresrom the night-levelbe-

causehe mainswing f the curvess confinedo thedaytime Part II);

day and nightare roughly eparatedy vertical inesdrawnat 6h and

18 . L can be described s a lunar semidiurnalwave with maxima about

sevenhoursafter the Moon's ransits; his waveappearsonly in daytime


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GEOMAGNETIC TIDES, HU-.4NCA I'0 273

.. Z

SLlOdAtDS ,NI t


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GE03 AGNETIC TIDES, HUANCA YO

275

In.o, c.o. 0

W,O; c.o ,*) '..

JAISAnY 19, 9$

(n=o; C.O./]

APPARENT MOON; $ClLE 0 PER DIVISION


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276

J. BARTELS AA'D H. F. JOHNSTON Wo,-. 45, No. al

FIG. --FOR SUNSPOt-MIN/,IUM ; DE,OARFURESFROA. IGHt-LEVEL, MAGNETIC HOR/ZONlrAL FORCE. HUA'CAYO, SELECIrED

OUIErDAYS ItHCO/VSPlCUOUSEOMAG.MEt/CIDESNJANUARYHOWINGNVERSIOEEft 70gOHrCf r/OAL

EFFECt WItH ,.GE Ill) OF A,EAN A400N i,'V HOURS--UNDERSCORED FOR DArE GIVEN WHICH ALSO tYPlF ES

--SUNSPOt-NUMBER, C--MAGNEtIC CHARACtER-FIGURE; UPPER AND LOWER TRANSITS FOR I EAN MOON '&FOR

APPARENr MOON; SCALE 201' PER DIVISION


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GEOM'AGNETIC TIDES, HUANCA YO 2??

than about eight days in January were available from which to choose

each of the eight records eproduced n the center of Figure 1. These

types f curvesre hereforeotvery. are; hiscanalso eseenrom he

fact that two of the contrasting pairs of magnetogramsshown are for

Januarydaysonly a quarter-monthpart, namely,A and B1, and

C and D.

TXBLZ 2--Eight selectedbig-L-days"n Figure 1

[Agev of meanMoon,distances-p) rommeanperigee--bothn hours t Greenwich

noon---unspot-number , international magnetic character-figureCs

magnification-factorsand X in (vS*-{-Xl.*)i

Day

Date

(s-p) R C

A

Jan. 19, 1923

Jan 27, 1923

Jan. 13, 933

Jan. 29, 1935

Sunspot-minimum

. 15.7

13. 7.0 42

9. 18.7 11

4.1

3.5

3.4

4.0

el

Jan. 25, 1928

Jan. 22, 1937

Jan. 8, 939

Jan. 15, I939

Sunspot-maximum

163

86

9.

2.9

2.4

3.1

2.4

Table 2 gives details for the eight selected ecords, ncluding he

magnification-factorsand k whichapproximatehe observed urve

by the combination$*q-XL*) f the averageariations* and .* for

sunspot-minimumnd sunspot-maximum 6). These parameters

a and X shouldbe accepted s rather hypotheticalestimates or reasons

given n 6. Table3 gives he average ariations.

5. The model-setsf "big-l-days"

All 527Januarymagnetograms,923-39, eresystematicallyearched

for "big-[.4iays,"nd womodel-setsereselectedor sunspot-minimum

(Figs.2 and 3) and sunspot-maximumFigs.4 and 5) for prospective

useasguiding xamplesn measuringhree-hour-rangendices . Each

set consists f 12 daytime curves,namely,one curve or each age v of

the Moon to the nearest hour; in contrast to Figure 1, no distinction

was madebetween he ages or (v:e12) hours,becausehe main tidal

effectsshouldbe the same for v or (:12), if distance-effects re neg-

lected.

Table4 gives ata for the daysof the model-sets. he average un-

spot-numbersor the days n the two setsare R--3 andR--84. Only

quietdayswereselectedaverage =0.34); thus he number f days

(withC 1.0,say) romwhich o select ach urvewasnothigherhan 6.

Since,or mostagesv, severalmagnetogramsere ound ust as typical

as the one inallyselectedor the model-set,t canbe said hat out of


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278 J'. BARTELS AND H. F. JOH.NSTON [vol.. 45, No, sl

T,m,E --Averageolar nd unardailyvariations* andL* n horizontalntensity, uancayo,

January, 1923-1939, or sunspot-minimura nd sunspot-raaximum,n depart-

turesof hourlymeans r night.level

[L* is given or a quarter-monthnly, or ages --0 to 5 hours;L* (v+12) = L*(v);L*(+6)--

l*(vq-18)=-l*(); d=average departure, without regard to sign,

for daytime 06 to 18h; dx is explained n 7]

Mean

solar

hour

h h

00-0 1

01-02

02-03

03-04

04-05

05-06

06-O7

07-08

08-09

09-10

10-11

11-12

12-13

13-14

14-15

15-16

16-17

17-18

18-19

19-20

20-2I

21-22

22-23

23-24

d

dz

Sunspot-minimum,

L* orv=

3 4

Sunspot-maximum, R = 83

for'=

........

2

- 26

+ 84. 3 - 81

- 80

- 4

-1-t-159

,1,299

-t-398

,1,405

+294

-t-138

- 4

-80

- 77

- 26

- 2

+ 5

168

91

every ten days in Januaryat leastone shows _as prominently s the

magnetogramsn Figures2 to 5.

The "observed"curvesare tracingsof the magnetograms or daytime,

6 to 18 . A straight line has been drawn connectinghe consecutive

night-levelsslightly lopingbecause f non-cyclic ariationsand the

scale s indicated by parallel lines drawn for intervals of 20-,;sincethe

scale-valueof the //-variometer, for instrumental reasons 7], changes

acrosshe magnetogram,hesehorizontal inesare morenarrowlyspaced

near the top of the curves,as is plainly visible n the curve or v= 2 in

Figure 2. The instrumental ffectof temperature-changes,ess han

1% is negligible.

Each "observed"curve has been approximatedby a "computed'"

curve,marked (S+L), which s a superpositionf two curvesmarked


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GEO.tL..GNETIC TIDES, H ANCA YO

279

i.a.t; c,o e)

(R=Z; C,,O.)

('e= 05; C,

(,e; .0.)


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280 J. BARTELS AND H. F. JOHNSTON Wot.. 4s, No. 3

.)

)

J ]I-'4

1929

c.o.o)

29,1956

ir=da; oo.?)

1f--5 ll=i?

' i

FIG. ,5 -FOR SUNSPOF-A4'AXIMUM;EPARTURESROMNIGHT-LEVEL, AG_NEFICORi2'ONX'ALORCE, UANCAP'O,ELECTED

QUIET DAIS W/IN CONSPICUOUS EOMAGNEF'iC'IDES IN JANUARY $HOH/ING NVERSIONLEFT I'0 RIGHT'OF TIDAL

EFFECT'II'H GE'U,} FMEAN4OONNHOURS--UNDERSCOREDOR ATE IVEN HICHlSO YPIFIES12 2,}i

R=SUNSPOT-NUMBER, =MAGNE'7'IC HARACTER-FIGURE;PPERAND LOWER 7'RANSI'S:% OR MEAN MOON, t FOR


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282 J. BARTELS AND H. F. JOHNSTON Win.. 4s, No. 31

half of the anomalisticmonth n which k is systematically reater s

centered t (s-p)=4 hours,after perigee;but there are exactly 12 days

in each of the halves centered at (s-p)--4 and 16 hours. Thus, the

influenceof the lunar distanceon the occurrence f "big-I_-days" eems

negligible, although, if more days had been selected, the influence of

(s-p) would probably have becomeclearer.

Transit-times, nearest to noon, of the mean Moon (calculated from

v) and of the apparentMoon (taken from the "AmericanEphemeris nd

Nautical Almanac")are given n Table 4 and indicated n Figures2 to 5.

They differ in one case by 0.76 hour--46 minutes in time; these differ-

ences will be referred to in Part II.

5. An attempt o separate and on individualdays

The main features of a daily variation in //can be consideredas

given by a row of the 12 hourly means,or even six two-hour means,

between 6 h and 18h, expressed s departures rom the night-level and

correctedfor non-cyclicvariation. Such rows may express he "ob-

served" curve on a particular day, or the average variations $* and

k* for a specifiedime of the yearand a certainaverage unspot-number.

Table 3 gives $* and [.* for January; how these values have been

computedwill be described n Part II. Only the "heart" of that Table--

the 12 values from 6 h to 18 in each column--will be used to determine

the "magnification-factors" and X expressinghe observedcurve as

CS+L), with $=$*, L=XL*.

As an example, consider the sunspot-minimumday January 29,

1935, with ,--20, pictured in Figure 2; the hourly departures rom the

night-level are given in Table 5. The first two lines repeat, from Table

T,3r.E mSeparationvS*+XL*) or dailyvariation n January 9, 1935,horizontalntensity,

ttuancayo; departuresrom night-level

Standard 75 west meridian hour

Variation

06-07 07-08 08-09 09-I0 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18

S* +13.2 +28.3 +56.9 +84.8 +97.8 +91.6+69.9'+37.0 + 9.8 - 4.5 - 7.0- 3.1

L*(---8) - 2.8- 6.2-10.0-13.2-12.4- 9.5- 1.8+ 8.3 +15.9 +18.1 +13.61+ 6 '

Obs. 0 - 5 + 2 +30 +73 +90 +92 +91 +79 +55 +29 + 4

1.2 $* +16 +34 +69 +103 +118 +111 :+85 +45 +12 - 5 - 8 - 5

4.00 L* -11 -25 -40 -53 -50 -38 - 7 +33 +64 +72 +54 +25

Synthetic + 5 + 9 +29 +50 +68 +73 +78 +78 +76 +67 +46 +20

Residual - 5 -14 -27 -20 + 5 +17 +14 +13 .+ 3 -12 --17 -16

3, $* and L* (20)=L* (8)= -L* (2). The third linegives he "observed"

departures. The approximationof the "observed"departuresby a

"synthetic" curve (a$*+XL*) requires he determination of, and X so

that this equation s, as nearly as possible, alid for eachhourly interval.

In other words, 12 simultaneousequations for and X must be solved,

namely: (+13.2,-2.8X)=0; (+28.3-6.2X)=-5; (+56.9- 0.0X)=

+2; etc. The method of least squares yields =1.21, X=4.00. The

"synthetic" curve 1.21 $*+4.00 L* fits the "observed" curve well, as


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GEOMAGNETIC TIDES, HUANCA 2}'0 283

shown by the "residuals," that is, the differences observed" minus

"synthetic." The relative mportance f the superposedariations an

be judged by the averageabsolutedepartures rom the night-level,

whichare given n the last four columns f Table 4, namely, for the day

consideredn Table 5, 46 for "observed,"50-r for $*, 39/for Xk*, and

14-/for the "residuals."

Becauseof the limitations to be discussed n 7, the least-square

methodswere abbreviated in most cases, ither by using wo-hour means

or by condensinghe 12 hourlyequationsnto two linearcombinations

favorable for the separationof r and X. In our example, by adding the

four hourly departures rom 8 h to 12h, and the five hourly departures

from 13h to 18h, the generalequation (r$*+Xk*)=observed] yields the

combinations, xpressedn the unit

(331.1a-45.1X) = + 195

(31.5cr+62.2X) = +258

with the solution r= 1.08, X = 3.60. These values give also quite a good

fit, with the average esidual aisedonly to 15-/from 14-,as in the case

of the orthodox least-squaremethod.

Both Figure 2 and Table 5 showclearly that a part proportional o

L* is contained n the observedcurve, and that it would be hopeless o

regardhe observedurve s a multiple f S* alone;n fact, the least-

sq,are ethods--giving.78*as he estitof he bservedurvey

alone--leaws residualsaveraging as much as 34-.

7. Uncertainties f this separation

The separation f $ and L in the daily variations n singledayspro-

videsan approacho an accurate limination f L, if S is to be usedas

a measureof solar wave-radiation W; fu,'thermore, the high variability

of L in itself deservesstudy. Therefore, the reliability of the computa-

tions described n 6 will be briefly discussed.

(a) As shown n detail in Table 5, for eachday with age v, the two

magnification-factorsr and X give an equation

Observed epartureromnight-level [a$*q-X[.*(v) residual] (7.1)

for eachhourly nterval,so hat thereare 12 such quationsor eachday.

The magnitude f each erm s given, n the last ourcolumns f Table4,

by the average epartures,akenwithout egard o sign. Onemightbe.

inclinedo judge he successf the separation y the relativemagnitude

of the average esiduals ompared ith the smalleroneof the averages

for ,$* and X[.*--mostly the latter. This reasonings, however,only

correctas far as the representationf the observed urveby the "syn-

thetic" curve (r$*-l-XL*) is considered, ut does not reflect on the

reliabilityof the determination f a and X individually. Indeed, f, for

a particulargeof theMoon,L* would ave hesame hape s$* (say,

L*=q$*, with q positive r negative),heapproximationf the observed

curvewoulddetermine aq-qX)only, and all pairsof values r and X

with the samevalue (a+qX) would furnish he samesyntheticcurve.

In general, emay ind, oreach gev of theMoon, parameter

satisfyinghe 12equationsone or eachhourly nterval)

L* ,) = q(v)S* L*.t.,) (?.2)


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284

.1. BART, ELS AND H'. F. JOHNSTON [Vot,. 45, No. 31

by the condition hat the sum of the squaresof the 12 departuresde-

fining L*.u(u)shouldbe as smallas possible.This least-squaredjust-

ment (as well as other discussionsn this section)can be pictured n an

orthogonalcoordinate-systemf 12 dimensions, ne for each hourly

interval; eachrow in Table 5 defines he end-pointof a vector from the

origin; and equation (7.2) means hat the vector L*(u) is represented s

the sum of two vectors,one, q(v)$*, in the direction of $*, the other,

L*z(v), perpendicular o $*. With (7.2), (7.1) becomes

Observeddeparture= [(cr Xq)$*+ XL*. v) +residual] (7.3)

The averagedeparturesd for L*() and d. for L*.(v) are given in the

last two lines of Table 3, for v=0 to 5 hours; they are the same for the

ages v, (v+6), (v+12), and (v+18) hours. The values d. show that

the "significant" lunar variation L*.u(v) is greatest for v=0, 1, or 2

hours, and smallest for v=4 or 5 hours; these values confirm the im-

press[on,with respect o the degreeof similarity, afforded by the curves

marked $ and L in Figures 2 to 5.

The values Xda., which are the average departures for XL*.(u),

shouldbe comparedwith the average residuals. In one case (sunspot-

minimum, v=4) the average residual (=12-r) exceeds Xd. (=10).

Table 6 summarizes he magnitudesof the average departures given in

the last four columns of Table 4, and supplements them by those for

XL*.(v); the values for sunspot-minimumand maximum and for three

consecutive values of v have been combined. For the "unfavorable"

agesmaround v or (v-12)=4 and 10 hoursrathe average departures

for the significant unar variation XL* are reduced o nearly one-half

those or XL,but remain still about twice as largeas the average esiduals.

TXBLE 6-A veragedepartures rom night-level or

the days given in Table 4

Age

u or (u-- 12)

hours

O, 1,2

3, 4, 5

6, 7, 8

9, i0, 11

All

Ob-

served

69

76

75

44

66

o'S*

54

46

68

,57

$6

XL*

40

35

30

32

34

XL*.s.

36

10

27

17

2,5

Resid-

ual

15

8

10

9

10

(b) The average variations S* and [.* have been chosen rather

rigidly. The restriction to two average sunspot-numbersR is not

seriousas far as the approximation is concerned,because he change of

$* and l* with R is more an increase n the amplitude, and lessa deforma-

tion of the shapeof the daily curve; more detailed choiceof S* and [.*

with regard to R would therefore have changed , and X, but not the

residuals. The restriction of L* to round values of , the neglect of

the lunar-distance effects, and of the seasonalchangesof S* and L*

between January 1 and 31 may have causedpart of the residuals;but

since the computations were mainly illustrative it was not thought

worth while to go into greater detail.


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286 J. BARTELS AND H. F. JOttNSTON p,'o.. 45, No. 31

.f(t) = (S+L+ D)

which may be studied separately. The intensity of the disturbance-

variation D depends on that of the solar corpuscular radiation P.

On quiet days D consistsof a recovery, a non-cyclic variation which

can be easily eliminated. What "quiet days" are, in the sense that D

is negligible,dependson the station, on the magnetic element, and on

the problem considered. For Huancayo, near the equator. days with

C


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GEOMAGNETIC TIDES, HU'ANCA YO 287

It is practical o start with smallgroups, o study the variability of the

averages, nd then combine hem into sufficiently arge groupswhich

show he systematicdependence learly enough. Groups ormed in this

paperhavebeendescrild n 2. Results or smallgroupswill be

mostlyexpressedn tabular form in orderto allow the combinationwith

resu ts rom years to comeor from other observations,while results or

large groups,as the basis or discussingystematiceatures,will be

giveng.'raphically.

9. Chambers'sunisolarvariations, nd Chapman's hase-law

For a definite station, magneticelement, season,and sunspot-number,

Chambers I 1] expressed., in solar time t, for any age v, by

k(v) = [a(t) cos 2v+b(t) sin 2v] (9.1)

so that [.(v) is determined by _(0).--= (t) at New 5Ioon, and k(3)

at one-eighth hase. If a(t) and b(t) are expressedn hourly means, .

is determined by 48 values. With ordinary harmonic analysis, the sine-

waves or periodsof 24, 12, 8, and 6 hours n a(t) and b(t) would be de-

terminedby 2X2X4= I6 parameters. [If a(t) and b(t) contain constant

terms, there would be two more parameters. This possibility has not

beenconsidered efore;it postulates hat the ordinary daily meanschange

with the age v of the Moon in a semimonthlywave, which will be dis-

cussed n Part I I.]

Chapmanreduced he number of parameters o eight, namely, four

amplitudes and four phase-angles, in his phase-law, xpressed

either in mean lunar time r or mean solar time t (in the latter case neg-

lecting he small changeof the age in the courseof the day)

[_()---2: c. sin [++(-2)1=2:c. sin [t+-2v] (9.2)

(9.2) is a special aseof (9.1), expressinghe two setsof harmonic oeffi-

cients or a(t) and b(t) by oneset of coefficients, . and ., namely,

a(t) =2:c sin (vt+,), b(t)=-Zc cos tq-) (9.3)

In this paper,we shallusemainly the moregeneral xpression9.1); but

the validity of (9.2) will also be tested (Part II), and a new general

expressionor _(v)will be introduced.

10. Two waysof computing

Consider a lunar variation k expressed n hourly means. Each

hourlymeandependshenon variableswhichstand n the relation2(4);

the possiblendependent airingsare (t, r), (t, v), or (r, v), all of which

have been used. We choose t, v), as in (9.1).

The analysisof a functionof two variables rom observeddata may

be begunby regarding nevariableas a parameter kept constant) nd

studying he variation with the other variable. Which of the two

variables s chosenas parameter in this initial step of the analysis,

should be irrelevant for the final result. In the caseof k(t, v), the two

possibilitiesre to regard as parameter nd to study .(v), the variation

with v (fixed-hourmethod escribedn [1] as van der Stok's method),or

to regardv as a parameter nd to study .(t), the variationwith t (fixed-


20/40

288 J'. BARTELS AND H. F. JOHNSTON [Vo.. 45, No. al

agemethod escribedn [1] as Broun'smethod). Both ways have been

used by various authors in geophysicalproblems,also their equiva-

lentswith [_expressedy (*, v) or (t, ,). The fixed-agemethod has been

furthestdevelopedn a recentpaper by S. Chapmanand J. C. P. Miller

[3]; the fixed-hourmethod has been recently applied by M. Bossolasco

[18],J. Egedal 19],and W. J. Rooney 20].

In geomagnetism,he two methodsare not equally effectivewith

regard o the elimination f irregular isturbance-effects.he fixed-age

methodseemspreferable, ecauset removes rom k(t) automatically

the averageeffect D of the storm-timevariation (ring-current ffect)

on the daily means,and can easily be modified o eliminate the non-

cyclicvariation oo. With the fixed-hourmethod,k() appears s a

lunarsemimonthly ave (9.1) in eachhourlymean; he fluctuations f

the daily meanscausedby D will, however, ntroduceconsiderable

semimonthly avesof a randomnature,whichwill maskk(v) unless

they are eliminated, or instance,by expressinghe originalhourly

means s departuresrom the daily meanor from the night-level.

11. Use of ranges

The statistical isadvantagef the fixed-hourmethod ies n the fact

that it judges (v) by the changesn hourlymeansromday to day

which are superposedn much greater rregularchanges--whilehe

fixed-age ethod(t) judges by the changesrom hour o hour--from

which he superposedargechangesue o $ canbeeliminated ecausef

their regularity. The weakness f the fixed-hourmethodappears

particularly ronouncedn the alliedproblem f computinghe atmos-

pheric idesfrom readings f barometric ressuret ground-stations,

becausehe greatpressure-wavesonnected ith the weather-changes

introduce emimonthlyaves f muchgreater mplitudeshan [_,even

at equatorial tations.Onlyhourlyor bihourlyeadingsf recording

barographs,reatedby the fixed-age ethod,eemeddequate.With

the usual hree barometer-readingser day at climatologicaltations,

it appearedopelesso attempt calculationy thefixed-hour ethod

until it was realized 12]that the effectof these ong-period eather-

wavescan be neutralizedby studying .(v) in the changesf pressure

between wo successiveeadings. The experience ained n these

pressure-studiesed o theextractionf L fromgeomagneticaily anges,

originallyntroducedor thepurposef measuring.

12. The ranges andAs

Various indsof dailyranges avebeenused o indicatehe magni-

tudeof the dailycurve(t) as analyzedn (8.1). In the nternational

scheme f the "numerical haracter"whichoperated rom 1930 o 1939,

RH was the difference etween he highestand lowest nstantaneous

values f H in a Greenwichay. R is--aswas ntended--muchffected

by D; andso s,althoughess, range erivedrom hehighestnd

lowest ourlymeans.Fora range-definitionuitableor measuringhe

intensityf $ and atherndependentf D, heshape f theS-curven

Huancayosuggestshe ise fthenoon-levelfH overtsnight-level

[13],so hat the range s expressedy

A =(averagerom h o 14h)minusaveragerom h o 5h) (12.1)


21/40

GEOMAGNETIC TIDES, HUANCA YO 289

taken with sign. More exactly, A is the excess f the averageH for

(9hto 14h)overthe night-level orrectedor non-cycl.ichange, sshown

in the reproductionof a magnetogram n Figure 6. The non-cyclic

change s eliminated by measuringA from a straight line connecting

consecutiveaverages for (0 h to 5h). Since the centers of the five-hour

intervals are at 2h.5 and l lb.5, nine hours apart--incidentally sym-

metrical to 7 , or Greenwich noon--the numerical correction for non-

cyclicchange s easily made. If N' and N" are the valuesof the con-

secutivenight-levels 0h to 5h), the correctednight-level s [N'+(9/24)

(N"-N')]; this is subtracted rom the noon-level 9h to 14 ) to give A.

Of course, f one of the night-levels is highly disturbed--for instance,

if a storm beginningafter 14 depresses "--this correctionmust not be

applied schematically; herefore, n suchcases, he non-cycliccorrection

is estimated. This summary procedure is justified since the effect

of the non-cyclic changeon A amounts to only a few per cent for days

with C< 1.2. To show he order of magnitude: In the (rather disturbed)

monthsJanuary 1926 and January 1938, the averagenon-cycliccorrec-

tion [that is, (9/'24) (N"-N'), taken without regard to sign] was 4.0-/,

to be comparedwith an averageA -- 8,; in the (quiet) monthsJanuary

1931 and 932, the correspondingigureswere 2.0, and 76.

Such rangesA were computed or all days with C< 1.2 for the whole

available series,March 1922 to October 1939. For the months December,

January, February, in the sunspot-maximumTable 1), A was also

computed or the days with C between1.2 and 2.0.

Daily rangesof H which are obtained by eliminating the effect of

L from A (see 18) will be called As, since they can be used to measure

the intensity of $.

13. Absence f appr,ciabledisturbance-effectsn the rangesA

Averageswere formed for the two kinds of ranges2 and Rr for

groups f dayswith increasingmagnetic haracter-figure , as explained

in 2 (10), and collected n Table 7. The first row refers to sunspot-

TAm.7---Daily rangesA (of five-hour means) and Rr (of instantaneous alues), horizontal

intensity,Huancayo; veragesor four or ive classes f daysqo,q, 2, qaand d, in

the orderof increasing nternationalcharacter-figure

Averages or months

Equinoxes, 1931-34

Equinoxes, 1936-39

Dec, Jan, Feb, 1935-39

RangesA on days

I ' I I " I '

21 86f901 61...

142l 140113911421...

125125[32[

RangesR on days Ratios (Rr/A) on days

q___0q__q__q__}_ad_.[.[.l[ --'_f.]

517]T47131...11.4711.5911.

188121 204[ 2231... It.3211.4311.4711.57

17379[05[19[


22/40

290

BARTELS AND H. F. JOHNSTON [VOL. 4s, No. 31

minimum, the secondand third rows to sunspot-maximum. Of course,

R>A always; but while R increases distinctly with C, A remains

practically constant, at least for the first four groupsq0 o q3,which are

subdivisions of all days with C< .2. The small differences, in each

row, between the average A for groups q0 to q are of the order of the

probable errors and therefore statistically not significant.

A systematicchangeof A with C might have been expected or two

reasons: ,4 would increase with C, if the solar wave-radiation W should

increaseparallel with the increaseof solar corpuscular adiation P indi-

cated by C; but A would decrease ith C as soon as the solar daily dis-

turbance variation $) becomes ppreciable,because he shape of the

$D-curvemakes the range A as defined by (12.1) negative. An inde-

pendentestimateof/1 for $) is possible or the material on (from the

five internationalquiet days per month, our q0-days) nd ($-55) (from

the five international disturbed days per month) for Huancayo H in

the average or the sunspot-maximum ears 1926-29 R =69) as published

in graphical form [ 3]; the average character-figures or these inter-

national quiet and disturbed days are about C--0.1 and C--1.5. The

average ranges A are contained n Table 8. The decreaseof A due to

St) appears n Table 7 between he qa-days nd the d-days.

TA.BLE 8Average ranges A of fie-hour means in horizona ntensity,

Huancayo sunspot-maximum,years 1926-29

December- Equi- June-

Item solstice noxes solstice

Internationaldisturbeddays ($a-l-$r)) 107.1 116.8 90.9

Internationaluietays SSD 119.1 130.3 93.4

isturbance daily variation --12.0 - 13.5 -2.5

There is a theoretical possibility that the two influences of solar

corpuscular adiation P on the range ,4 could both be strong, but just

neutralized in the averages' The increaseof A due to a simultaneous

increase of W correlated with an increase in P, might just counter-

balance. he decreaseof ,4 due to the direct effect of P, namely, $.

This is, however, quite unlikely, because he effect, on ,4, of $x), even

for rather high disturbance, s not so large as the changeof A with the

sunspot-cycle. It seems hereforesafe to conclude hat, as far as changes

of the range A on days with C < 1.2 in the courseof the month are con-

cerned,an increaseof P has no appreciableeffect on $. This fact makes

,4 more valuable as a measure for solar wave-radiation W. It further

obviates, in computationsof [., the necessity o considersubdivisions

(groupsq0,q, q, qa)of the quiet days since t is certain hat $ is sys-

tematically the same for all these days with C< 1.2. That this holds

also for 1_will not be tested here directly, but may be inferred from the

results of 17.

14. Computation f the unar semimonthlywaves n the rangesA

The daily ranges 4 on days with C < 1.2 are written in 12 columns,

for or (-12)=0, 1, 2,... 11 hours, as indicated in the sample


23/40


9-.Computationf unarsemimonthlyaven daily anges , Huancayo, ebruary,

sunspot-group ax .for dayswith C less han 1.2

in hours

Febru ............

aryn 0 1 2 3 4 5 6 7 9 10 11

ear 12 13 14 15 16 17 18 19 2 21 22 23

1927 $2p 125 109 142 220p 185 .......... 94 90p 144 134

927 146 134p 124 ill 130 145p ..... 132 135 120 129 .....

1930 .......... 150 154 ..... 192 124p 109 SS 131 96p. ....

1930 ............... 130 145 130 105 65p 44 96 961

1937 163p168 191 148 187p 76 ..... 119 2sp'ii' 206

1937 211 24 192p ..... 223 213 ............... 118p 52 .....

1938 111 161 185 148p ......................... 112 58

1938 166 133 175 134 152 149p 110 58 78 96 85 139

1939 33p 154 131 160 146 ..... 179 1f5 134 ........... 76

1939 127 112 ..... 191 170 143 158 108p 75 94 100 106

A_v_rage 39 147 151 155 165 168 142 104 92 109 110 131

Wave +5 +13 +17 +21 +31 +34 +8 -30 -42 -25 -24 -3

Result: AverageA =134-y;semimonthly ave=(0.7vcos2vq-31.17 in 2)

--31. 'y sin (2q- )

Table 9. For pairs of successivendisturbed ayswith equal ntegral

valuesof [see 2 (4)], the averageA has been entered (markedp

in Table9); but for the monthswith 30 or 31 days,wheredayswith the

same occurat the beginning nd at the end of the month, he ranges

A for these aysareentered ingly. The blankspacesndicate, f course,

disturbed ays,or the fewdays or which he magnetogramasdefective.

The averagesre formedwithoutegard o the lettersp; it is true that

this gives heseaverage angesor successiveairsof daysnot the

weightheydeserve,ut,becausef quasi-persistencen S, this elative

weight, omparedithA fromsingle ays,s certainlyesshan2, and

experiencehowedhat a moredetailedreatmentwouldhardly m-

prove he accuracy f the semi-monthlyaves.

The lineof averagesor the 12 columnss thenharmonicallynalyzed

to yield the semi-monthlywave

aa cos2v+ba sin 2v=ca sin (2v+) (14.1)

If the analyzed alueswereaveragesor intervals f 15 in u,or 30 n

2v--because is rounded off to the nearest hour--the effect of this

"smoothing"hould e corrected,n a, ha, and ca, by applyinghe

factor (r/12)/sin r/12)]=1.012. Actually,sincepairsof successive

undisturbedayswith equalu havebeengivenhalf weight, his factor

shouldbe somewhat maller;we choose--bya somewhatengthyargu-

ment whichneednot be given here---the actor .010, whichwill be

correcto onepart in 1,000,even hought depends n the percentage

of pairs n whichonlyoneday is undisturbed,ndwill thereforeary

somewhatwith the sunspot-cycle.

The coefficientsa, ha,aregiven n Table10-A. The average oeffi-

cients1922-39 or each month,marked All", are means f the coeffi-

dents or the oursunspot-groups,eightedroportionalo the number

of monthsn the groupssee able1), for instance,or January, ll--


24/40


25/40

GEOMAGNETIC TIDES, HUANCAYO

293

on Table 10-A, give averaged nd smoothedesults. Figure7 illustrates

the main results n two schematic urves, or sunspot-maximumnd

sunspot-minimum,howing ow the rangeA wouldvary, in the course

of the year, under he combinednfluence f [. and of the season n $,

in a year supposedo beginwith newIoon on January1. The vertical

lines ndicate he epochs f new oon. The changen amplitudec.of

the semimonthlywave from its greatestvalue n January o its smallest

value n June s quitemarked; n addition, here s a shift n phase,with

the crest of the wave occurringearlier (with respect o new and full

Moon) in June than in January. The seasonal hange f A due to $

is obviously ifferent n character rom the seasonal hange f [ (see

16, a).

Tam..10--Lunarsemimonthlyavesa. cos +ba sin 2v--ca (2,+u,)] n daily ranges in

hori.ontalorce at ttuancayo,and averagesor As---daily ranges orrectedor lunar

influence nd R .--Ziirich elati,,e unspot-numbers

T,m.s 10-A--Original resultsor monthly roups

group aa ba at bt at ba aa bt at ba ax ba

Januaryebruaryarch April1 May June

Min, in 0.1 -142 +124 - 22 +189 q- 18 q- 66 q- 66 q-164 q- 23 'q- 62 q- 55 q- 42

Min., in 0.1 -52 q-136 --13 +295 q- 52 q-141 q- 67 q-113 q-61 q-61 q- 47 -16

Maxa,n .1--xs9-192-6q-160- 7 -178-26 -137- 9q-8 - 2 - 6

ax4, n 0.1, -26 q-229 q- 7 q-311 'q- 94 +202 q- 42 q- 43 q- 29 q- 42 q- 87 - 16

All, in 0.17 -172 +173 -- 5 +237 q- 60 q-145 + 39 +14I q- 4 [+ 53 q- 55 + 9

All, cx, in 0.17 244 . ..... 237 ...... 157 ...... 146 ...... 67 56 ......

All, ., in . ..... 315 ...... 359 ...... 22 ...... 15 ...... 38 ...... 81

July August September October November December

Mint, in 0.17 + 84 q- 67 [q- 59 q- 71 + 50 +104 q- 46 + 50 q- 10 .q-144 --47 +125

Min., in 0.17 - 4 q- 40 + 29 +111 q- 8 q- 66 + 87 + 28 -79 q-48 --61 +203

.Max=,in 0.1 + 53 + 23 + 37 + 24 +118 + 18 + 14 +142 + 2 +166 --63 +217

Max,in 0.17 + 21 + 66 i+ 46 + 50 +110 + 59 + 3 +203 + 79 +199 -- 31 '+243

All, in 0.17 + 40 + 51 i-}-44 + 64 q- 72 + 64 + 36 +108 + 13 +148 --49 +195

All, ca, in 0.1, 65 ...... 78 ...... 97 ...... 114 ...... 149 ...... 201 ......

All, a., in . ..... 38 ...... 35 ...... 48 ...... 18 ...... 5 ...... 346

. TxL, 10-BResults for seasorugroups,derived rom Table 10-A

Sunspot-

group

Mint

Min:

Max:

Max

All

Dmb-oati -I June'solstice

1 o

[660 1541341 25 8[648 82

20 I 775 78 I 343 I 28 1980 i 102 32 26 [ 734 59

67 [1082 1941342 I 57 [1219 [ 128 21

22 1041068 8

43

34

55

52

46

. ,


26/40

294 J. BARTELS AND H. F. JOHNSTON [vm.. 4s, No. S]

TXBLE 10.C--Smoothedesultsor monthlygroups,obtainedrom Table 10-A

[The smoothed oefficientsA`,b,t for groupswereobtained s follows: Feb."-- (January+

2 February+March)/4,etc.; "Min"--(2 Min+Min:)/3; "Min:"--- (Min+

Min:+ Max:)/3; "Max:" -- (Min+ Max:+Max0/3; "Max(" =

(Max+2 Max0/3]

SuBspot- CA aA CA. aA CA. aA CA. aA CA. a.A CA aA

group

0.17 o 0. v o 0 Iv o 0. v * 0. v * 0. Iv ,

" " "june"

anuary "February... March" "April ... May"

"Min " 70 334 170 350 40 11 122 24 88 33 65 49

"Min:" 193 335 179 352 155 12 122 20 80 33 56 52

"Max:" 232 336 219 354 188 14 130 20 72 38 49 63

"Maxd' 272 332 252 352 211 14 136 17 70 35 55 65

All 219 333 200 351 171 13 128 2 78 35 57 58

,

"July" "August... September... October... November... December"

"Min" 77 43 92 31 89 30 80 25 104 356 148 337

"Mind' 61 45 77 38 85 38 89 23 124 354 171 337

"Max:" 49 43 70 39 91 40 14 19 155 359 203 340

"Maxd' 56 50 73 52 104 41 157 17 199 5 228 343

All 63 46 79 39 94 37 114 20 150 1 189 340

,

Txm,. 10-D$moothed esultsor seasonal roups, btainedrom Table 10-B

[The coefficients A`,bAwere smoothed etweensunspot-groups

as in Table 10-C, but the seasonswere kept separate]

$unspot

group

"Min:"

"Max:"

"Max("

All

December-solstice Equinoxes June-solstice

aA

--50

-54

--$5

-55

--53

0.1'r

+ 154

+167

+2OO

+225

+ 88

CA

162

176

207

232

195

342

342

345

346

344

0.1

+48

+48

+54

+57

+52

bA`

O. I'r

+ 93

+ 01

+119

+141

+114

CA,

0.1v

105

112

130

152

126

27

25

24

22

25

0.17

+48

+44

+41

+45

+45

O.lv

+56

+46

+38

+34

+44

CA

41

44

47

53

46

TB. 10-ERatios (1000cA,/As) or smoothedmplitudesA, Tables 0-C and 10-D) and sim-

ilarly smoothedairy rangesA s

Smoothedmonthly results Seas

Dec.-

Jan. Feb.Mar. Apr. May JuneJuly Aug. Sep. Oct. Nov. Dec.solst.

233 210 15" 43 119 102 118 121 102 93 145 224 232

222 188 153 127 96 78 84 92 88 92 148 214 210

219 192 159 116 75 58 56 70 78 97 149 205 201

223 194 158 108 65 57 57 65 80 118 169 200 96

224 189 155 122 86 72 78 85 87 105 158 209 206

Sunspot-

group

,.., ............

"Minz"

"Min:"

"Max:"

"Max4')

"All"

group

Equi- June-

nox sotst.

109 85

107 63

112 56

111 76


27/40

GEOMAGNETIC TIDES, HUANCAYO 295

15. ;Estimatesor scattering

(a) Supposehe seriesof observations ere nfinitely ong,and the

coefficientsa, b for the lunar semimonthly wave in A were calculated

for each individual calendar month. Collecting all the months of

Januarywith the samesunspot-number,ayR=0, we shouldind that

the coefficients ax, b) would not be identical for each month, but

would differ, by residual coefficients zSaA, Abe), say, from the total

average or all monthsof Januarywith R =0. This scatteringn the

results for the individual months can be pictured in a cloud of points

[14] n the harmonic ial, in which he wave n each ndividualmonth s

representedsa point (endof a vector)with aa, btas plane ectangular

coordinates, upward,batowardthe right; and the amountof scattering

can be measuredby the probable error-radius p0 of the cloud, defined

as 0.833 times the standard deviation, or square-root of the variance

which is the average(/Xa=a+/xb',t). The scattering for average waves

derived from n months (of January, with R=0) is then pictured by a

point-cloudnwhichheprobablerror-radiuss p0/'n). Theprobable

error-circle, describedwith the probable-error radius around the long-

time average,divides, under ideal conditions, he cloud of points into

two halves,with equal number of points insideand outside;but in this

paper, he probable-errors just a constant ractionof the standard

deviation.

(b) Since he error determination s only of secondarymportance,

and since the observational material is limited, the orthodox scheme ust

describedwas replacedby the following, ess aborious rocedure.

(e) The numberof different "elementary"probable-erroradii p0

(measuringhe irregularscatteringof semimonthly ine-waves alcu-

lated from individual months) was reduced to six, namely, one for all

the months in each of the three four-monthly seasons, for sunspot-

minimum and sunspot-maximum.

(/) Each difference etweenan "original"wave (in Tables 10-A

or 10-B) and a "smoothed"wave (in Tables 10-C and 0-D) is used o

estimate 0.

(,) We assume,or simplicity, hat the cloudsof points or indi-

vidual months with equal R are circular. As far as the evidencegoes,

it seems o indicate that, if the cloudsare ellipsoidal, heir major axis

may lie alongthe averagevector,that is, the amplitudes cattermore

than the phase-angles;his would modify the considerationsn the

followingsubsectionse)and (f).

(c) The actualcalculations basedon the fact that %, as defined

above, s a fixed multiple of the variance,or squareof the standard

deviation. If x, y, . . . , are independent tatisticalvariableswith the

probable rror , p, . . . , then the square f the probable rror o

linear combination

jx+ky+ . . . ( 5.1)

is

i =p=={- =p=,-[- . . , (15.2)

this "law of propagation f errors"holds lso or two-dimensionalrrors,

as in the caseof the clouds n the harmonicdial, and implies he ordinary

law hat arithmetical veragesf n values,withequal ndividual robable

errors p, have the probableerror


28/40

296 J. BART. ELS AND H. F. JOHNSTON loL. 4s, No. 31

Let for the moment the four pairs of original coefficientsa., b,) in

Table 10-4 for the four sunspot-groupsn each month referred to as

Jan, Jan, Jan, an, etc., while the smoothedvalues may "Jan",

"Jan", etc. For instance, in the unit

Jan (- 14.2, + 12.4), according o Table 10-A

"Jan" (-7.4, +15.3), according o the values c, in Table 10-C

The difference s (-6.8, -2.9). In a long-timeaverage, his differen

("Janx"Jan) is to be expectedmuch smaller. Neglecting his small

systematic difference, the square amplitude, (6.8+2.9 ) = 7.4 , can

consideredas an estimate for the variance of the difference, and 0.833 X

7.4 as an estimate for the probable error.

Now, according o the double smoothingprocedureexplained in

2 (7) and (11), or in Table 10-C

"Jan" [(2 Dec+4 Jan+2 Feb+Dec+2 Jan+Feb)/12] (15.4)

("Jans"Jan) [(2 Dec-8 Jan+2 Feb+Dec+2 Jan+Feb)/12]

Each individualmonthly resulthas the sameprobableerror oo,according

to our assumption (); according to (15.3), with n=5 for the Min-

groups,and n=3 for the Min-groups (seeTable 1), the group-averages

haveprobablerrors 0/ andoo/, resctively. Therefore,15.2)

gives for the square of the probable error of (15.5)

o%x +o0 (82/5)/144] = o%/2.96 6)

For this value, we had the estimate (0.833X7.4); therefore, p0=

(2.96 X 0.833 X 7.4) = 18.2 . Other differences between smoothed and

original values yield further estimates for 0, and from the average e0

for veral such determinations, the six values for p0 in Table 11 are

obtained.

Txr 11robabl, error-radii o for lunar semimthty waves n Huancayo

H computedrom indid months unit, microgauss=0.1,)

Group

Sunspot-minimum

Sunspot-maximum

November to

February

136

66

. Equinoxes

......

112

117

May to

August

69

67

These estimates for o0 are likely to exceed the correct values by a

few per cent because the differences between the original and the

smoothed values will contain a systematic part while our calculation

considered hese differencesas due only to irregular scattering. This

keepsus on the safe side f we judge the reliability of the resultsby the

probable error-radii o given in Table 2, which are deduced from the

values o0 n Table 11 by applying (15.2) and (15.3). These radii have


29/40

GEOMA GiVETIC TIDES, HUA NCA YO 29?

TxsLz 12--Probable-error adii for the averageunar semidiurnalwaves n Tables O-A to 10-D

Group

Original

,

Min, in 0.1

Min., in 0.

Max, in 0.

.Max, in 0.

All, in 0.1,

Smoothed

'Min", in 0.

"Min", in 0.1r

"Max", in 0.1

'Maxd', in 0.

All, in 0.

Jan.]Feb.

ar.]Apr.

ay

June

uly

Aug.

Sop.

Oct.[Nov.[Dec.

Results in Table 10-A

61 61 50 50 31 31 3 31 ,50 50 61 61

79 79 56 56 34 34 34 34 56 56 79 79

83 83 58 58 33 33 33 33 58 58 83 83

74 74 52 52 30 30 30 30 52 52 74 74

37 37 27 27 16 16 16 16 27 27 37 37

Results in Table 10-C

29 29 25 22 16 14 14 16 22 25 29 29

26 25 21 18 13 2 12 13 18 21 25 26

28 27 21 19 13 2 12 13 19 21 27 28

35 33 26 23 16 14 14 16 23 26 33 35

2 20 16 14 10 9 9 10 14 16 20 21

Dec.- Equi- June-

mlst. nox solst.

ResultsinTable 10-B

30 25 5

39 28 17

42 2O 17

37 26 15

18 14 8

Resultsin Table 10-D

24 19 12

22 16 10

23 16 9

28 20 11

18 14 8

alsobeen used o construct he probable-error ircles n Figure 9. That

the probable rrors for January nd December, ebruaryand Novem-

ber, etc., appearequal n Table 12 is, of course, consequencef our

restriction of p0 to six values.

The physical ignificancef Table 11 will be discussedn 15 (b).

(d) The ratio of the amplitude x to the probable rror-radius is

an index for the "reliability" of the wave determination. These ratios

are of the following order'

For the monthly means, our originalsunspot-groupsTable 10-A)

c, =3 to 4 for November o March, but as low as in june, Maxs; for

the smoothedmonthly means, our sunspot-groupsTable 0-C)

lies between about 7 for December, January, and February, and 4

for June.

For the total monthly means, (row marked All), cao lies between

3 and 6 for the original esults Table 10-A), and between and 10 for

the smoothedvalues (Table 10-C).

For seasonal verages,our sunspot-groups,riginalresults Table

10-B),c. o liesbetween and 6; for smoothedesults Table 10-D),

between 5 and 9.

For total seasonal verages,inally, the ratios c,t/p are as high as 1

for the December-solstice,for the equinoxes, nd 8 for the June-solstice.

(e) We nowconsiderome onsequencesf ourassumption-), hat

the cloud of points is circular.

In general,f o is the probable rror-radiusor a circular lud of

points,heprobablerroro for theone-dimensionalistributionf the

projectionsf the pointson onedirections only

= 0.573p (15.7)

This follows, or normaldistributions,rom he facts, hat the standard

deviation of the one-dimensional roections s equal to the two-

dimensionaltandard eviation f the circular loud ivided y

and hat the factors y which hestandard eviations ustbemultiplied


30/40

298 J. BARTELS AND H. F. JOHNSTON Vo. , No. 31

to obtain the probable errors are 0.8396 for os and 0.6745 for o. This

gives he factor 0x/o)--[0.6745/(0.8326X/)]=0.573, that is, (15.7).

With ( 5.7), provided that the error-radius o is small comparedwith

the amplitude ca, the probable error in the phase-angleaa can be esti-

mated as 0.573 (o/ca) in angular measure; for instance, 3.0 for the

total averagewave in the December-solstice.With (15.2), we can further

decide whether some of the changes n the phase-angleaa in Table 10-B

are real or accidental, and find that the seasonal increase of aa from

January to June is doubtlesslysystematic, while the slight changes n

ea with the sunspot-cycle annot be regarded as statistically significant.

(f) We may, therefore, regard the average phase-angle aa as a

constant for each month or season, ndependent of the sunspot-cycle,

and we can use (15.7) also to estimate the probableerrors n the changes

of the amplitudes ca with the sunspot-number. The probable errors of

the amplitudesare pictured as vertical bars n Figure 10, which represent


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GEOMAGNETIC TIDES, HUANCA YO

299

the (smoothed) mplitudes n Table 10-C as functions f the averageAs.

The significance f the increaseof c. with the sunspot-number an be

judged rom the followingvalueswhichshow he algebraicncrease f

ca or typical values n Tables 10-Band 1O-C, ollowedby - the probable

(one-dimensional)rror of this increaseunit 0. -r):

December-solstice: (Max4-Min) = (251-154) = +97 - 27

Equinoxes: (Max4 - Min ) = (164 106) = + 58 =21

June-solstice: (Max4-Min) = (58-82)=-24 - 12

"January": ("Max"-"Mini") = (272- 70) = + 102 26

"June": ("Max4"-"Min") = (55-65) = - 10=11

These values, and Figure 10--in which the vertical bars drawn for

' " "Max"

Minx and . indicate the limits within which the true value of

cz can be expected to fall with the probability one-half--show that

the increasesof ca with the sunspot-cycle n the six months October to

March are significant,because hey are three to four times as large as

their probable errors. The increaseof ca is most conspicuousor the

December-solsticewhen the increase (Max4-Min)--(251-154)/--+

97,, for an increaseof R from 9 to 93, reacheshalf of cx= 95- for the

total average wave for that season. As to the computed decreaseof

OBSVArO'A,Vt,,SS,Ai &rrOs , o L S'M/-


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300 J. BARTELS AND H. F. JOHNSTON [vm.. 4s. No. 31

cA from sunspot-minimum o sunspot-maximum n the months May to

August, t appears ess eliable;but it can be said that an increase n the

june solstice nalogouso that for the December-solsticeby (63/2)

is definitely excludedby the observations.

16. Discussionof the semimonthlywaves

The facts in Tables 10 to 12, illustrated in Figures 7 to 10, show the

following main featuresof [_ n Huancayo H on days with C


33/40


(Table11) vary, n the course f the year,more ike cx than ike

This is shown n the following three lines, which express 0, c, and A

in sucharbitrary units that the average or the three seasonss 100 for

each parameter.

Parameter December- Equi- June-

solstice noxes solstice

Probable error-radius 0 136 103 61

Amplitudect 152 08 49

RangeAs 98 16 86

This result is of interest. If, namely, 0 were mainly causedby a day-

to-day ariability f $, it wouldmean hat thisS-variabilityhould e

less han half as big in June as in December--an nferencemade un-

tenableby a direct est [ 19 (c)Vii. An alternativepossibilitys that

the day-to-dayvariability of the two daily variations,expresseds

multiplesf their espectiveanges.say, s for $, and2c for L) is so

much reateror [_ han or $, that 0 sgoverned oreby thevariability

of [. than by that of $, and the decreasef c fromDecembero June s

reflected n a decreaseof p0. This latter assumptionseemsconsistent

with occurrence f the "big-l_-days" nd the discussionsn 4-7,

and will be considered gain in 19 (c) VII.

There s yet another spect. It had been hought hat [. couldbe

studiedmost successfullyn thosegeomagnetic bservationsn which

it ispronounced.his ed o concentratingttentionn he hreemonths

December,anuary, nd February. The ratio (o/c.),which s a good

expressionor he elativeeliabilityf thewave omputedrom single

month, s then about (15.1,/22.7,)=0.66.But for the equinoctial

months,his atio sonlyslightly igher, amely, bout 11.45/12.8/)

0.89. Becausef (15.3) he number f equinoctial onths equiredor

the same elative reliability of the semimonthly ave is therefore nly

(0.89/0.66)=1.82imes he number f months ecember,anuary,

andFebruary.Or, for thesame umber f months,ndcomparedith

months round anuary,he relativeprobablerror-radius,/c,,will

be 0.89 .66) 1.35 imes iggerorequinoctialonthsnd 1.08/0.66)

= 1.64 imesbiggeror monthsMay to August.The statisticalalue

of months round une or studies f [. in H at Huancayos therefore

not somuch nferior o that of months round anuaryas the decrease

of cxwouldndicate ecausehisdecreasen cx spartlycounterbalanced

by a paralleldecreasen 0.

(c) Changeith hesunspot.ycle--Ashownn 15 )and (f),

only heamplitudes, not hephasesx,changeignificantlyith he

sunspot-number. Figure 0showshat, n thesixmonthsctobero

March,when x s large,t increasesithR just ikeAs. FromTable

10-C, veragesorc were ormedorthefivemonthsNovember"o

"March";hey regiven erewithcorrespondingveragesorR andAs.

Sunspot-groupR As ca (cx/As)

"Min" 13 76'r 14.6 0.192

"Min" 32 89/ 16.4q, 0.184

"Mina" 60 108 19.9 0.184

"Min" 83 124 23.2 0.187

All 4,8.5 99.9 8.6'r 0.186


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35/40

GEOMAGNTICTIDES, HUANCA YO 303

In order o arriveat As, we must liminate z. This s doneby

assuming,or each calendarmonth,a lunar semimonthlyine-wave

c sSn2v+,.4) as follows:The phase-angless are those iven n the

last ineofTable10-A, ndependentf thesunspot-cycle.heamplitude

c isassumedo bea function f theaveragemplkude for themonth;

this functions gSven,or eachmonthseparately,n FSgure 0. The

actualcomputationunsas n the followingxample,or January 923:

The monthly verages for the monthsDecem'mr922, anuary nd

February923, re57,73,and82,,which ivesor hesmoothedverage

A for "January 923" he value71-. FromFigure10 we read,by a

slight extrapolation, the value 16.87 for c.. in order to eliminate the

effectof smoothing,e multiply hisvalueby the ratio (24.4,y/2 .9y)

of theoriginalo the smoothed onthlymeansor January, sgiven n

Tables 0-Aand 10-C, ndobtain heround alue 'f197 or c. There-

fore, he daily valuesAz of the wave 19rsin (2v+315) are subtracted

from hedailyvalues f A in january1923; heresults a rangewhich s

freed rom he averagenfluencez of 1_.Thiscorrectedaily anges

calledAs and s assumedo expresshe intensityof S.

The monthly veragesf A andAs do not differsystematically,ut

in each ndividualcalendarmonth he differenceAs-A) may differ

from zerobecausehe calendarmonth s longer han the lunarmonth,

and becausehe disturbed aysare omitted. Thus,on the average,

for January 1937, (A-A)=(153-158)=-Sy. But these individual

monthlydifferencesAs-A) are generally maller,as shownby their

standarddeviations rom their averagevalue zero, namely, for the

monthsDecember,anuary, nd February, when _ s large)2.4,y, nd

for the monthsMay to July (when _ s small) 0.6%

19. Influence f on thevariabilityof A and As

The cases f abnormallyarge . discussedn 4 to 7 make t prob-

able that the corrected angesAs may show esidual unar influences,

becausehe correction escribedn 18 eliminates, o to say, only

the normal .. SinceAs is proposeds a measureor the intensity f

$, we will discuss ow much [_still affectsAs; furthermore, he vari-

ability of [_ tself is of interest. The followingdiscussions restricted

to typical cases and is not intended to exhaust the rather intricate

problem of the joint variability of S and [..

(a) A curious istakend ts esson--Theollowing ayof studying

thevariability f [. suggeststself: In theaverageor December,anuary,

and February, at sunspot-minimum,he average 2t and the semi-

monthly wave in .4 is given by

A = (As+Az) =[71,+18y sin (2v+341)]

The maximumof the semimonthlywave occursat v--(109/2), or

approximately v4 hours. This means that, for all days with v or

(v-12)=4 and 10 hours,at the maximum nd minimumof the wave,

the.average is 893, nd53% espectively.f, now,Az on these ays

vanes much about its mean value 18% .4 on these days will be occa-

sionally much higher or much lower and will fluctuate more than for

ages 1 and 7 hours, when A z is small. A seeminglyunbiased est of

this consideration as made by selecting,n each calendarmonth, the


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304

J. BARTELS AND H. F. JOHNSTON [VOL. 45, No. 31

three days with the three highest rangesA, and the three days with the

three lowest ranges A; from all these ranges (144 in number) for the 24

calendar months the lunar semimonthly sine-wave was computed in the

usual manner ( 14), and gave

73-+41- sin (2v+339 )

While the phase-angle aa =339 is practically equal to that given in

(19.1) as derived from all days, the amplitude c.=41v is more than

twice as large as the ordinary amplitude cA= 18/given there;according

to (19.2) A varies with the Moon's age, on these selecteddays, between

(73-41)/and (73+41)/, or in the astonishingratio 1.0 to 3.6.

At first sight one might be tempted to draw the conclusion hat the

intensity of [. is more than doubled on these six selecteddays per month

with the three highest and the three lowest rangesA, thus confirming

our suspicion hat the variability of [_ s large. This would, however,

be a curious mistake which deservesmention as a standard example of

erroneous interpretation of biased sampling, of the same nature as a

mistakerecognized arlier[16]. There is, namely,another,quite different,

interpretation which, although likewise schematic, is nearer the truth-

We may assume that in the equation A =Asq-A., day by day, AL

dependsrigidly on v as expressed n (19.1), without any deviation, while

the whole irregular variability of A is due to As which fluctuates around

its average value 71v according to a frequency-distribution hat is in-

dependent of v. The ranges A, for days with v or (v--12)=4 hours

xvould hen fluctuate around their average value 89% and the rangesA

for days with v or (v-12)=10 hours would fluctuate around their

average value 53?. The three highest angesper month would preferably

occur on days with v or (v- 12) around 4:hours, and their average value,

say, Amax,would be higher than 89- by an amount depending on the

standard deviation of As and the relative number of days selected;

likewise, the three lowest ranges per month would preferably occur on

days with v or (.-12) around 10 hours, and their average value, say

Am, would be lower than 53v.

This is confirmed by the following somewhat condenseddata of our

example which comprised24 calendar months'

Three ranges Number of dayswith v or (v- 12) hours Total AverageA

per month 0, 1, 2 3, 4, 5 6, 7, 8 9, 10, 11

With highestA 16 42 11 3 72 1107

With lowest A 16 6 14 36 72 36

Furthermore, the deviations hA of the daily rangesA from the average

As for each calendar month were formed; for instance, A =60 for the

day January 1, 1923, he averageAs for the month of January 1923, s

75, therefore /XA =-15 for that day. These values have been

plotted at the left in Figure 11, distributed ccordingo

hours.

The average emimonthlyine-waves, ith the parametersc.,; a),

for the months n sunspot-minimum, re formed from these daily de-

parturesAA as (18.2; 329) for December, 20.8?;320) for January,

(21.3; 355) for February; n order o eliminate he seasonal hangen

phase-angle,he dayswith v for Februarywerecombinedn the graph


37/40


38/40

306 I. BARTELS AND H. F. JOHNSTON LYon.4s, No. 3]

3.3% Therefore, the influenceof on the highest and lowest valuesof

AAs seemsso small that it cannot be detected with certainty, at least

not in this way; evidencespeaking for a slight residual influence will be

discussed n the next section (c), and then there are, of course, those

lunar influenceswhich originate in tidal terms other than M2 (to be

discussedn Part II), and those features of [_ n the hoursafter 14 which

are so conspicuous n the "big-L-days," but are not expressed n the

range .4.

(c) The oint variabilityof $ and L--The right half of Figure 1 gives

the impression hat the dots scatter more for the agesaround --4 hours

than for those around v= 10 hours. This is confirmed by the standard

deviations, (/kAs) for As, namely,

for v or (v-12), in hours,=0, 1, 2 3, 4, 5 6, 7, 8 9, 10, 11 All

a (AAs) = 9.3 22.0 20.2 18.3 20.0

The excess 22.0-18.3)=3.7-r is small, but significant; since, namely,

the standard deviation of is (e/x/2n), where n (= 153) is the number

of independent observations, he probable error of 3.7, can be estimated

as 1.1-r leaving a margin even if n were effectively reduced by quasi-

persistence n AAs. This result throws some light on the variability of

A =As+An, or, if we call A0 the monthly average of A or As, on the

day-to-day variability of /xA = (A -A0) = (AAs-4-A0. Consider a

few ideal cases'

(I) If An never deviates from exactly ca sin (2v+ax), its variance

is ,(A.)--da/2, and

r(A = a'(As) +ca/2 (19.4)

In our example, with (As) =20.0q,, ;t=20.1T [after (19.3)], this gives

a(A0=14.1qr, and (A)=24.5% This consequence s purely mathe-

matical, due to the least-square method of determining cx, and a test

by computing (A) directly would be nothing more than a check on

the harmonic analysis, but not on the hypothesis.

(II) Suppose,next, that Ax.=cv sin (2+ax) is only an approxima-

tion to the actual lunar range, which may be (A.+ 8.). Then, the solar

range is [A-(A.+&.)], and its deviation from the average A0 is 8S--

[A-A0- (A-4-80]. In other words,

/A, = (A --A,--A o) - ( 8s-4- 0

(19.5)

(III) If 8sand &. were statistically independent,and if the standard

deviations , (8s) and (8,.) were independentof v, (19.5) would give

d(/XA.) =

(19.6)

likewise ndependentof v; this contradicts he observation,as we have

seen above, since , (/XAs) is 22.0 for v=4 hours, and only 18.3q or

v = 10 hours.

(IV) If we keepthe assumptionhat 8sand 8,.are independent, nd

that, (/s) doesnot dependon , then the decrease, rom v=4 to = 10,

of ,2(/xAs) from 484qa to 335? would be wholly due to a decreaseof

,a(;x.). Even if , (Sn) were zero for v= 10 hours, this would give, for

v=4 hours "(&)= 149a, and, (&.)= 12.2%making a (8,.) nearly equal

to, (An)= 14.1-r. This estimate or, (&.) appears ather high, because

it reaches 61 per cent of ca.


39/40

GEOMAGNETIC TIDES, HUANCA YO

307

(V) We therefore est the hypothesis hat 5; and [. vary propor-

tionally,an assumption hich wouldbe natural n the dynamo-theory,

if the currents or $ and [. flow n the same onosphericayer. Suppose

t,/'A,=f (/sA0) = (19.7)

with a constant ratiof. Then

/xA= ( + aD =,[(A 0,"7)AI (9.8)

( 9.9)

If r () is supposedndependent f r, the ratio of our values, (Aq) for

r=4 and10wouldive 22.0/18.3)l[(71/T)+20],:'([71/f)-2] ,r

about f= 1/3. This is certainly too small, since, with , (s

20.0 '71.0=0.28, (19.7) would give a (/L,'AL) nly 0.09, obviouslycon-

tradicting he existence f the "big-[_-days" hich, n any case,would

suggest aluesof f exceeding nity. Or, put differently' If the vari-

ability of (S+[.) shouldvary with v proportionalo (A0+AL), we would

expect he standarddeviations (/xAs) for r=4 and 10 hours o stand

in the ratio [(71+20)/(71-20)]--(1/0.56); the values for

for v=4 and 10 actually found from the observationsshow, in fact, a

difference in the right direction, but the ratio is only (22.0:'18.3)--

(1/0.83). This points to the conclusionhat to a considerablextent

$ and [. do not vary in unison, but independently. The actual conditions

will be a compromisebetween the extreme hypotheses IV) and (V).

(VI) We compare he followingvalues for sunspot-minimum. For

June, average A 0 60, c = 4.9,, r (/XA) 15.67, r (AA s) = 15.27;

for the months December to February, average A0---71% cx---20.17,

(AA) = 25.4, (/xAs) = 20.0%

The relative scattering of the corrected anges,r (/XAs),/Ao is 0.25

for June, 0.28 for December o February, not much different,but con-

sistentwith the view that the smalleraverage ntensity of [. in June, as

indicatedby c,, s accompanied y a smaller absolute)variability of [.,

expressedn a reduction f (/kAs)/A 0.

(VII) The standard deviation of /xAs, at sunspot-minimum, as

been found above as = 20.0, in the months December to February,

and =15.2, in June. In Table 11 ( 15), the probableerrorsof the

lunar semimonthly wave from single months were found as - 3.67

and 6.9%respectively. n the caseof random cattering 15],harmonic

analysisof sets of n equidistantordinateswith standarddeviation

yieldsa root-mean-squareine-wave mplitude 2r/Mn), for any fre-

quenc_y,r a cloudof pointswith probablerror-radius=(0.833X

2a/v/n). Insertingour values or and , we computen=6.0 around

January,n-- 3.4 for June, or the "equivalent umberof random aily

valuesper month," o be compared ith the actualnumberof daysused

permonth,about25. This gives25/n--about 4 and 2for the "equiva-

lent number of repetitions"of daily values ZxAs, indicating he con-

servation endency n ,4 from day to day as expressedn semimonthly

waves. This number s higheraroundJanuary,when [. is large,possibly

becausehe variability of I_ s then also argeand quasi-persistent.

(VIII) Of the 607 days pictured in Figure 11, not one has a range

As double hat of the normal range; n other words, AAs neverexceeds


40/40

308 J. BARTELS AND H. F. JOHNSTON [vor..45, No. sl

A0. While $ thus never appearsdoubled, he "big-L-days"make it

necessaryo assumehis for [_.

(d) ;Effect f thevariabilityof l on theuseof As for measuringW--.

The preceding iscussionupplementshe impressionained rom the

"big-[.-days"hat [_varies, o a largeextent, ndependentlyf $. That

the day-to-day ariabilityof [. is greater han that of $not in absolute

units,but relative to the average ariations _*and S*---appearsn the

"big-l_-days" ostclearly n the afternoon ours n which$ is small.

From 9 to 1 h, when$ is large, he irregular luctuations f [_add only

little to those of $. Since these are the hours from which A is derived,

it appearshat the residualluctuationsf [_do not seriouslyndanger

the useof As as a measure or the intensity of $, even in those months

when[. is large. Especially,f As will be determined rom a numberof

observatories21], the residual nfluences f [_on the combined verage

of the As will probablynot exceed few per cent of the standarddevia-

tion of this proposed ay-to-day measureof solar wave-radiationW.

[3]

[41

[51

[6]

[71

[81

[91

[01

[1

[2]

[31

[4]

[tSl

[7]

References

[1] S.ChapmanndJ. Barrels, eomagnetism,vols., xviii+ 1049+77pp.,Oxford

University Press,1940. (The lunar magneticvariationsare discussedn

chapters , 8, 19, 20, and 23.)

[2] Terrestrialmagnetism nd electricity,edited by J. A. Fleming, (Physes of

the Earth, vol. VIII), xii+794 pp., McGraw-Hill Book Company, New

York and London, 1939.

S. Chapman nd J. C. P. Miller, Mon. Not. R. Astr. Sot., Geophys.Sup.,

4, 649-669 (1940).

J. Barrels ndH. F. Johnston, rans.Amer.Geophys. nion1940,273-287,940.

J. Bartelsand G. Fanselau,Geophysikalischerondalmanach, s. Geophysik,

$, 311-328 (1937) [partly translated n Terr. Mag., 43, 55-158 (1938)].

J. Barrelsand G. Fanselau,Geophysikalischeondtafeln 1850-1975. Abh.

Geophys.nst. PotsdamNo. 2, 44 pp. (1938).

Ref. [1], p. 86f.

A. T. Doodson,Proc. R. Soc.,A, 100, 305-329 (1921).

J. Bartels,Gezeitenkraefte,n Handbuchder Geophysik,1, 309-339,Gebr.

Borntraeger,Berlin (1932).

Ref. [1], p. 679.

Ref. [1], Chapter 8.

J. Barrels,Beitr. Geophysik, 4, 56-75 (1938).

J. Barrels nd H. F. Johnston, err. Mag., 44, 455-469 1939).

Harmonic dial and cloudsof point are described n [1], Chapter 16, or in J.

Bartels, Terr. Mag., 37, 291-302 (1932).

Ref. [ ], _Chapter 6.

Ref. [1], {}8. 8.

Ad. Schmidt, Archiv des Erdmagnetismus,Heft 7; appearedas Berlin, Abh

met. Inst., 9, No. 1 [Ver6ff. No. 357] (1928).

[18] M. Bossolascond J. Egeda, Terr. Mag., 42, 123-126 1937).

[19] J. Egedal,Terr. Mag., 42, 179-181 1937)and43, 89 (1938).

[20] W. J. Rooney,Terr. Mag., 43, 07-118 (1938).

[21] J. Barrels,Solar adiationand geomagnetism,err. Mag., 45, 339-343 1940).

DEPARTMENTOF TERRESTRIALMAGNETISM

CARNEGIE NSTITUTIONOF WASHINGTON,

Washington,D. C.

bartels johnston

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