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OCR for page 245
CH 4\PTER ,XVI
THE VARIATION OF LATITUDE
By. D. L.\MBrCRT, FR.\NK SCHLESINGER AND E. N\. BROWN *
U. S. Coast and Geodetic Surve?J, Yale Observatory, Yale Vniversit?J
PRELIMIN A:RY CONSIDERATIONS
The astronomical latitude of a place is the angle between the vertical
of the place and the plane o:l the equator; or, id: we wish to bring into
the definition the earth's axis and its poles, we may say that the colatitude
(complement of the latitude or ninety degrees nitrous the latitude) is
equal to the angle between the vertical of the place and the axis of the
earth. It must therefore be remembered that the latitude may be changed
either by a change in the direction of the earth's axis or by a change
in the direction of the vertical.
Let us discuss the latter briefly, chiefly in order to put it aside at the
start and to concentrate our attention mainly on the former. A real
chafe in the direction of the vertical at an observatory may be due to
~ movement of the ground on which the observatory rests with respect
to the body of the earth, thus brin<,in`, the observatory directly under
a different member of the set of verticals that help to define the earth's
field of gravitational force. Such cha.n~,es would occur on a large scale
if TVegener~s hypothesis of Doating continents and continental migration
were true. This hypothesis is discussed elsewhere ~ in this series of ~eo-
physical reports and will not be further alluded to here, except in
connection with possible causes of secular displacements of the earth's
axis. It would be out of place here to discuss NVegener~s hypothesis, with
its attendant shifting of great masses through great distances, but move-
ments of the ground on a scale small as to both the amount of the motion
and the extent of the territory involved are known to occur in regions
subject to earthquakes.
Sudden movements of the earth's crust, both horizontal and vertical,
usually occur at tile times of major earthquakes, and, according to one
theory, these movexnents are preceded by slow creeping movements in
*After conferences among the three members of this committee, this chapter
was written chiefly by \N7. D. Lambert, some paragraphs being contributed by
Frank Schlesinger.
~ It is expected that a brief discussion of the Wegener hypothesis will be given
in a forthcoming bulletin of this series on the constitution of the interior of the
earth.
245
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246
FI CURE OF THE EAR TH
the contrary direction followed by a rebound at the time of the earth-
quake. The maximum known amount of these horizontal movements is
of the order of a few meters,' that is, a very few tenths. of a second of
arc at the most, but displacements of even this amount are, so far as is
known, confined to the immediate neighborhood of those faults, the move-
ment along which occasions the earthquake in question. So far as is
known, no important astronomical observatory is located so near a region
of seismic activity that any appreciable change in its latitude has been
clearly traceable to seismic disturbances. Nevertheless, this does not
imply that continuous and careful observations of latitude might not
be useful in detecting or disproving the existence of a suspected crustal
creep in regions of seismic activity.!
There is another way in which the direction of the vertical may change.
The earth's field of gravitational force, and with it the direction of the
vertical may be altered. The forces that produce the alteration may be
external or internal.
Obvious external forces are the tide-producin~ forces of the moon and
sun. These produce deflections of the vertical of the order of 0".01. The
subject is properly one for treatment under the general heading of Earth
Tides (see Chapter V), but since observations of latitude have been ana-
lyzed to bring out these changes of the vertical, the subject will later
be discussed briefly in connection with the variation of latitude. No other
external forces seem likely to produce any measurable effect.
Internal forces, on the other hand, have been su<,~,ested as capable of
producing a rearran~,ement of matter within the earth and hence a change
in the earth's field and in the direction of the vertical of a given place.
It must be confessed, however, that results from forces of this sort do
:not seem very probable, particularly changes. as large and as rapid as have
sometimes bleed su~,~,ested as perhaps due to this cause. In the interior of
the earth below the depth of isostatic compensation the equilibrium is
* Much larger vertical movements of 100 or 200 meters are reported to have
occurred in Sagami Bay in connection with the Great Tokyo Earthquake of
September 1, 1923, but the reality of these has been questioned. Vertical dis-
placements on the shore of the bay did not greatly exceed two meters.
t another means of studying such a seismic creep is the surveying and re-
surveying at intervals of the suspected region by means of careful triangulation.
But such a triangulation has to start from a given base-line considered as in-
variable, and this invariability may be a matter of opinion. The farther a, given
point is from such a base-line the greater is the possibility of accumulated observa-
tional error through the intervening triangulation. In astronomical observations
for latitude each observatory is independent of every other, though this indepen-
dence does not obtain in the case of longitude and azimuth. However, granted an
invariable base-line not too far away, the accuracy obtainable from triangulation
exceeds that of astronomical determinations.
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VARIA TI ON OF LA Y'I TIDE
24r?
supposed to be nearly hydrostatic in character and the distribution of
density symmetrical about the earth's axis of rotation, so that practically
no effective ehan~,e is to be looked for in the deeper interior. In the earth's
crust, however, changes may and do occur. Erosion and deposition are
going on, mountains are foamed by uplifting and ocean beds by de-.
Session movements occur along faults; erosion and deposition, uplift
and subsidence are slow processes, the effects of which can scarcely be
expected to be perceptible within historic time; and though crystal move-
ments may be relatively rapid, a little ea.leulation will show that the
masses involved and the amount of the displacements are too small to
have more than an insignificant effect on the direction of the vertical
_,
within historic time.
The apparent, as distinguished from the real, direction of the vertical
may obviously be due to atmospheric refraction. Considerations of re-
fraction cannot be avoided in a discussion of the variation of latitude
but the subject will be treated latter in this chapter.
There remains then for present consideration only that part of the
variation of latitude due to the displacement of the earth's axis within
the body of the earth.* It is only changes with respect to the body of the
earth that need properly be considered. Changes of the direction of the
axis in space are included in precession and nutation. These cause changes
in the right ascension and declination of the stars, but not in the latitude
of a place. However, in astronomy the processes of determining latitude
and declination are often so intertwined that at any one place an error in
declination may be interpreted as a change in latitude. Hence it is im-
portant to have accurate declinations. If two observatories, however, are
using the same stars, and, on account of faulted declinations, there appears
to be a change of latitude, such a change would be common to the two
nb.c~rvn.t~rio~ Id the observers would thereLv be out on guard to look
a FJ IN ~ ~ Y ~ ~ V ~ ~ ~ ~ ~ ~ ~ ~ U ~ ,~ v ~- - 1~
to the declinations used.
HISTORICAL AND DESCRIPTIVE ACCOUNT
Aside from the vague suspicions that beset the early history of every
subject of this kind, the first consideration of variations of latitude
appeared in 1760 in Euler's mathematical discussion of the problem of
~ A displacement of this sort would change the field of centrifugal force due
to the rotation of the earth and hence the direction of the vertical, but if we
consider the change in latitude due to a displacement of the axis or of the pole as
a small quantity, then the change in latitude for an unyielding earth due to the
change in direction of the vertical is at most 1/588 of this small quantity and
may generally be neglected. For an elastically yielding earth the effect is only
a little greater The term " centrifugal force " is here used, as usual, to denote the
force required to balance the l~inetic reaction to rotation, or Muir.
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248
FI GURE OF THE EAR TH
the rotation of a ri~,icl boll,:- about its center of gravity.* Euler emphasized
the application of his theory to the case of the earth.
, The theory has
since been elaborately developed by later writers on the mechanics of
a rigid body and the results are letdown in ~ general way to most students
of the subject. The results can be stated without much mathematics,
the chief complication arising from the number of axes of rotation that
must be considered, along with their corresponding poles. It is usually
a little easier to look merely on the surface of the body and to consider
the motions of the poles rather than the axes; the latter may be con-
sidered as lines drawn through the center of gravity to the corresponding
poles.
TA:e have first the axis of figure. In a symmetrical body like a homo-
~eneous ellipsoid of' revolution it is the axis of revolution. Any body,
whatever its form or distribution of' clensity, has three axes about which
the moment of' inertia is a maximum or a minimum. Let us call that
axis about which the moment of inertia is greatest the axis of figure.
Then there is the axis of' instantaneous rotation; points on this axis
remain I'or the moment motionless. It is the direction of this axis that
determines with respect to! the body of the earth the pole of the celestial
sphere for the time being.
In an absolutely rigid body the positiol1 of the axis of figure is of
course fixed in the body, but is not necessarily fixed in space. There is,
however, an axis the direction of which, apart from external forces, is
fixed in space. For a body like the earth this axis coincides very nearly
with the instantaneous axis of rotation and for simplicity no distinction
will hereafter he made between them,. If, for instance, the axis of figure
and the instantaneous axis of rotation are separated by a small angle a,
the angle between the axis of' instantaneous rotation and the axis having
an invariable direction in space is about 300 a and so may be neglected.
For our purposes then the axis of instantaneous rotation may be con-
sidered invariable in space. If I'or any reason the axes and the poles of
instantaneous rotation and figure are not coincident, the pole of figure
will describe in space a closed orbit around the pole of instantaneous
rotation. On the other hand, with respect to the mass of the body the
pole of instantaneous rotation describes a like orbit about the pole of
figure.
~ Leonhard Euler. Theoria motus corporum solidorum seu rigidorum. Greifs-
wald, 1765. Chapter XII. A second edition was published in 1790. This work of
Euler's in the new edition of his works now in course of publication has not yet
been reached by the editors.
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7~\ oF [~
of
249
For definiteness let us Dmit ourselves to Me cam ~) taLc Sac origin
~ system of ~cot~n~l~r ages at Sac n01tb Sac ~ figure, Sac z-^is
being the Dais of Marc OD] the positive direchon of the -is lying
in the mcriJi~n of Grecnvicb. TO conformity aim the practice of the
Intcru~tion~1 [~tituic Scrvicc the positi~c direction of the praxis gill
be taken in the plane of Sac mcrTdi~n of 90° Scat. To an otsc~vcr looking
gown from the north, Sac Pole of inst~t~ncoos rotation dcsrritcs an
orbit i, the ncg~ti~c (coU~tcr-clockwisc) direction; abut is; from positive
p^ ~
If A, ~ and ~ arc the moments of inertia about the tbrec pTincip~1
arcs as ichnc] Stoic in ~sccnJing order of m~gnit~ic, ~ acing a{ COUTSC
the amount gout the -sib and if for Sac cab sac ~> same tat ~
~] ~ Pro aqua then the pub of the insts~t~neous pole about the pole
of rotation is ~ circle, icscrited in ~ sidemen Was, or gout ten
~o~ths.0 Ibis motion is ~ 7raa motion; sometimes Called Me mulcting
mutation.
If the Ji~c~gcncc Between the axis of Ognrc and the axis of inst~nt~nc-
ous rotation cc once to tccomc cst~Llishc] this Fulcri~U nutrition voUl]
tboorctic~lly continue indchnitcly, since no account is take of friction
in this acorn. Aatrono~crs dig not tbink it p~ot~blc Ant them could
romain any EUlcrinn nutrition of large amplitude continuing from the
ore ~ some c~t~atropbe in geologic~1 Sac and not act Jumpy out t
friction~I rcsist~nces, hut they dig think it possible abut cb~nges in the
Cam> CTUSt dUC to cs~tb~kea or caste mo~cmcn~ of over BOltS
might produce from time to Sac ~n EUlcri~n nutrition not too smog
to tic ictcCtc] fly c~rcfU1 otscrv~tion. Accordingly they looked for it.
casct as caky Pa 1S42, cx~mincd his otscr~tions for ~ tc~-months
oscillation, but could not kin] one. arc gas iolIo~c] fly PCtClS, ~{D;
gowning ~] ~e~comh, ~b of whom concluded thst their observations
coal] catkin no te~-months term Aim ~ coefficient as large as 0~.1.
Astronomers concluded that the term; if it existed ~t silt gas inscnsitle
each to the rebated methods then (1880) ~tt~ise] in practice astronomy
Ibis view was nnivers~ll~ ~cccptcd-unt] lSSS. As is very often the
case the otse^~tions Iron Ibid ~ variation of 1~titUde Bus first revealed
and announce] mere n~ded.~kcn vita an entirely different pn~pose in
view a~] without Any suspicion of ~ cb~nge in lattice. In lSS4 finery
~ See Helmert~ BObere Poodle, vol. II, p ~0, for the (morons solution, also
for off makers condensed with the m.~them~tic~1 tbeoly. For ~ cigar tre~tm~t
in [nglisb, see A. G. Webster, Ibe dynamics of particles and of diid, elastic and
Ouid bodies. Leipzig, 19~, a. 252. For diagrams see J. Clerk ~x~eN, OD
dynamic top. Cambridge University Press 1890, vol. I, a. 248.
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250
FI GURE OF THE EAR TH
of' Berlin, began a short series of observations for the purpose of testing
a method devised by him for determining the constant of aberration.
The.method employed involved the measurement of squall diff'erenees of
zenith distances in the manner devised by Talcott, of the Corps of
Engineers, United States Army. Although there was every reason to
believe that this method would lead to precise results, Kiistner found
that his separate values of the aberration constant agreed neither among
themselves nor with the best previous determinations. By a nice chain
of logic he was able to exclude one possible explanation after another
until only a latitude variation was left to account for these discrepan-
cies. Ibis next step was to examine other nearly contemporaneous series
of observations, and when he found that several puzzling, anomalies could
thus be accounted for, he no longer hesitated to announce ire 1888 a
variation of latitude. Ike stated that the latitudes of Gotha, Berlin,
and Pull~owa had each fluctuated to the extent of' a few tenths of a
second of arc within about one year.
This announcement at once awoke the liveliest interest, and the matter
was promptly talker up by the International Geodetic Association Obser-
vations were set on foot at Berlin, Prague, Stra.ssburg, and Potsdam,
and these showed art agreement that left few doubting the reality of
latitude variations. There were a few, however. The crucial test was
made in 1891 by the International Geodetic Association and the United
States Coast and Geodetic Survey. Observations for latitude were made
by the former at Berlin, Strassbur3, and Prague, and at TVaikil~i, in the
Sandwich Islands; and by the Coast and Geodetic Survey at R.ocl~ville
(in Maryland), San Francisco, and also at Waikiki. The last station
was especially important because its longitude is about 180° different
from European stations. Consequently, if the latitudes of the latter are
found to increase durir~<, a certain period then that of' Waikiki must be
expected to decrease simultaneously by the same amount. As this was
found to be the case, we may say that the two independent series off'
Marcuse and Preston, at TVai.kil~i, firmly established the fact that the
earth's axis of' figure is slowly revolving, around its axis of rotation.
Meanwhile S. C. Chandler, of Cambridge, NIa.ssachusetts, had already
begun his investigations upon the law of variation.: The results obtained
by him mark an epoch not only in this subject, but in the whole progress
of precise astronomy. :For he was able to trace variations of latitude as
far back as the time of Bradley, more than two hundred years previous,
* Chandler's articles are in the Astronomical Journal, beginni
(1892) and continuing with one or more in almost every volume
(1903).
rig with vol. 11
through vol. 23
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7~70\ 0~ [~
251
and to Bbow tat away of the discouraging discrepancies eDcoUntere]
since Ant Awe were due to Ions in Me l~tLuie.
Cb~iler~ tippet ~D~Dcement, ~] the most important, Bus to Sac
Fact that 811 rOccDt otacTY~ioDa oboe] ~ period ~ about 428 degas or
40 per Cant more Awn acre classic period Altbou~b Cb~nilcr) ovi-
dcnce on this point gas conclusive, his period gas beait~hDgl~ receive]
until Simon ~cwco~ poTDtc] cat (iD 1892) tab Fulcra Peony applied
only to s rigid toJ>; ~D] abut, if USCi i0I the c~tb, the theta must Bc
Zodiacs to talc ~cconDt of the cartes clOsticity. Ibc OXpl~D~tioD COD he
preseDte] VeTy clearly in ~ewcomt~s own MOTES: ~
~r Cbundlcr} discovery gives disc to the question ~bcther there can be any
defect in the theory ~bicb assigns 306 days as the time of rotation. Ibe objeo
of this paper is to point out that the is such ~ defect, Chicly, the fuGurc to take
account of the elasticity of the eat itself ad o{ Me mobility of the ocean.
Ibc (~thcmutic~1 throw of the rotation of ~ solid body, OD ~bicb conolu-
Jons b~bcrto rcc~ved brave been based, presuppose that the body is absolutely
rigid. As the carts and oceJu arc not absolutely rigid, he bane to inquire Better
their Oc~ihLity appreciably Hacks the conclusions. Abet it does can be Roan
Ace Imply Cow the following con~dCr~tioD:
Imaging the earn to be ~ bomogeneous Typhoid, entirely covered by an occaD
of .tbc same density limb itsclL It is then evident that, if the Bole mass he set
in uniform rotation monad ~ axis Whither, the ocean ~1 Bums the form
of AD obl~t~elipsoid of revolution, Bose Sadler axis coincides vita that of rot~-
hon. Back the ages ~ rotation Ed of figure ~1 be in poem coincidence under
3~_.
To apply ~ amber reasoning to the case of the cab, imagine tat the axis of
rotation is displaced by 0720 from that of greatest moment of inertia, Enrich I
sb~1 cut the axis of Marc. IBM, Cab ~ ocean o{ the she dcn~ as the earth,
as equator would be displaced by the same ~mount. Be OCC~D leycI Could
cb~ngc iD middle latitudes by about one inch at the maximum. But this change
Could bade for As ehect ~ corresponding cbunge in the axis of hours. As the ocean
covers only tbreeJour~s of the earn Be axis Bold be displaced by tbree~ou~bs
of the distance bet~cen the two ages here ocean and ebb of aqua denser. But
as the density of the ebb is some five times as great the actu~1 cage Could
be only one-fftb of thin It~oul] even be less tab one-Oftb, because the dis-
pl~cement of the ocean equator Could be resisted by the attraction of the Crib
LselE The exact amount of this resist~ncc can not he Cutely given, but I
trim the displ~cemcnt Could thercLy be reduced to one hula I tbercforc think
tab onc-fourtecntb Could be ao approximate csUm~tc of the displ~cemc~
of Tic axis of Hearst in consequence of Tic movement of the ocean. As
Cb~ndIcr~ period ~qnires ~ displacement of ~o-scven~s He off di~l~cc-
ment only accounts for one-fou~b of the difference.
~ On the dynamics of the e~rtb's rotation with respect to the v~ri~tiou of l~ti-
tude. So. Notices Roy. Astrou. Sock 32: 336 (1892).
~ about the Fact of 1bc yielding of the occult the free period of the lath
tude variation Could, according to the cshmute, be some 40 days less tab the
observed value.
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Representative terms from entire chapter:
proper motions
~2
FIGURE BE TIIE EAR TH
Else remainder is to be attributed to the elasticity of the earth itself. It is
evident that the flexure caused by the noncoinc.idence of the two axes tends to
distort the earth into a spheroid of the same form as that which the ocean
assumes, and thus to bring the two axes together.
We have now to Dow how this deformation of the earth changes the time of
revolution. Let us imagine ourselves to be looking down upon the North Pole
and let P be the actual mean pole of the earth when the two axes are in coinci-
dence and R the end of the axis of rotation. Then, in consequence of the rota.-
tion around R. the actual pole will be displaced to a certain point, P'. Now the
law of rotation of ~ is such that it constantly moves around the instantaneous
position of P' itself. In other words, the angular motion of R at each moment is
that which it would have if P' had remained at rest. Hence the angular motion as
seen from P is less than that from P' in the ratio of P'R: PR.
Belt as R rotates, P' continually changes its position and rotates also, remaining
on the straight line PR. Thus the time of revolution of R around P is increased
in the same ratio.
Newcomb's argument may be illustrated by Figure I. Inlet R and Ret
be " consecutive " positions of' the pole of rotation, separated by an in
R
~ ~r
,
R1
FIG. 1.-The various poles.
hnitesimal interval of time fit. The corresponding positions of the actual
pole of figure, as affected by the yieldings, of the earth, are P' and Pi' lying
on the lines joining R and R) with Pi, the point where the pole of figure
would be if' the pole of rotation were to coincide with P. Let ~ be the
angular velocity of' the point R about P' and ~ its angular velocity about
P. Then for the in-fi~itesimal distance RRI we have
RI?1 = (\PE ~ ad -{ = (
7~67~70\ O~ [~
~3
be rI] of thc centrifug~1 force ~lising from the non~oincidence of tbe
two poles ~D] tbe ~ttend~nt distortion oI the yiel~ing e~rtb. Sce GI14CI
3~. Tbe ch~nges 1~ tbe pos~ion of tbis ~oTe ~re JUe solel~ to ch~nges
In tbe distritUtion Qf m~tter ~itbi~ HD~ 08 thc e~tb ~p~t fro~ ~ny
stress ~ising f~m ~e r~tion. It is r~prescn~] t~ ~ i~ Bi~rc 1.
3 j Bet~een ~ ~nd ~ is tbe ~ctu~1 pole of f~ur~; 1~presented t~ [/ or ~/.
The displ~cc~ent oI thc ~ctU~1 pole ~y from tbe unJistu~he] pole
dUo to tbe sh~t centri1Ug~1 Iorce resul~ng from tbe noncoincidencc
of tbe pole of rot~tion vitb tbe unJisturted pole of ~gUre ~d to tbe
yielding of ~e e~rtb UDJer tbis centrifUg~1 force, ~ ~ielding ~t tenis
to Jeform tbe e~rtb so ~s to Cispl~c~ tbe ~ctU~1 pole of B~re ne~rer to
tbe pole of rot~hon.
Ibe equ~tions of motion of ~ Jo^y cb~n~ing tod~ b~e heen tre~ted
by v~rioUs ~utbors.# Convenient exposidons ~n] developme~ts of tbei~
wo~ vib bc Ioun] in v~rious ~ooks ~] tre~tises.I Ibere b~ve been
sever~1 tre~t~ents vitb speci~1 reference to ~e v~ri~tIon of l~tituie;
one oT the most recent ~nd tboroUgb is due to Scbv~.$ [or on~
pur~oses use vill Be ~Je of ~n ~rti~e t~ [~or ~ r~tb~ too lon~
to reprodUce bere; in v~icb be sbovs ty very gener~1 re~soning bov tbe
pr~lem of ~e motion of ~ yiel~ing tody m~ he reJUced to tb~t of ~
rigid one. [armor proves th~t for tbe ~tU~1 el~stic e~rtb we m~> sUtsti-
tUte ~ rigid e~ltb oI ~toUt tbe s~e YolUme tUt ~itb so mucb of its
eqU~tori~1 hUIge 1emo~e] ~s ~oul] he Jue to tbe ~iel~ing of tbe e~
under tbe entire centrifUg~1 Iorce of rot~tion. I1 Ibis voUld le~ve the
* J. Liouvibe. AddiLoDs ~ 1~ conD~ss~nce des temps pour 1839, or m~b~mu-
~quo~ puros ot ~ppliqu~os, vol. 3 (1858). Cyld~n: ~cchorchos sur 1~ rot~tion do 1
Terre; Actos do 1~ Soci6t~ roy~lo dcs 8ciences d'Ups~t vol. 8 (1871). Tbomson, W.:
Appendix C to D~r~in~ ~rticle on tbe I#inenco oF goolo*io~1 cb~Dgos on thc
E~b's ~is of rot~tion; ~iL Tr~ns. Roy. Soc. [ondon, pt. I, YO1. 167: 271 (1877);
or D~r~in's Scientito P~pers, vol. 3, p. 1.
iE J. Routb. AdYunced rigid dynumics; Stb ed~ p. 17. B. [~mbs Bigher
Dlecbunics (Cumbridge, [nglund, 1920), p. 171. F. Tisser~nd: ~6o~nique c~leste;
vol. II, Cb~p. X\~. F. R. H~mert: B6bere goodisie; vol. ~ p. 40S.
~ W. Scb~ed~r. Die ~c~egung der Drebucbse der J~sUscben Frde in Erdkhrper
und im R~u~e: A~ronomiscbe \~chichten, 203:101 (1010).
~ J. [~rmor. Ibe rel~tion of thc e~rtb'E free procession~1 nutution due to
its resktancc ug~iDSt hdu1 deform~tion, Proc Roy. Soc. London, sec. A, vol. 82: 89
(1909).
~ If tbe entire ceDtrFugu1 force ~ere removed, it ~ to be presumed tb~t tbe
e~rtb ~ould {~ f{~a yield pl~stic~lly un~ tske op ~ ncurly spb~ic~I form. This,
bo~evcr, is not ~h~t is ~C~Dt. Ibc yieldisg is to be t~[cn ~s el~stic, th~t i~ pro-
portion~I to tbe force urd not incre~ing ~itb thc ti~c, ~ccpt for thc yieldin) of
tbe oce~Dic ~ters ~lre~dy mentioned. The e~eot of the reurr~ngemeDt of tbe
ater ou tbe equ~Iori~1 bulke, or ~ore precisely on ~ ], ~ust be ~llo~ed for in
254
FIGURE OF THE EAR TH
earth less flattened, and since the differences of the moments C-A and
C-B serve to measure the flattening,, we must replace these differences by
something, smaller by about 3/10 of itself. Although the relative changes
in the differences of the moments are considerable, the change in any
one moment is relatively small when compared with the moment itself.
The smaller flattening of the equivalent rigid body corresponds to a
longer free period. The free period is thus about 3037~1-0.3~=433
mean solar days.
For a yielding as for an unyielding earth, a difference between the
values of A and B betrays itself in the elliptical form of the path of the
pole for the free motion. To dist.in~,uish the free motion of the actual
earth, with its fourteen-month period, from the theoretical Eulerian
ten-month motion we may call the former the C1~an~dlerian, motion. The
longer axis of the elliptical free path lies, for the Chandlerian as for
the Eulerian motion, in the plane containing the axis of the larger of the
two principal moments of inertia. In a hon~o~,eneous ellipsoid this axis
would correspond to the shortest equatorial radius.* A study of the
Char~dlerian motion may therefore be made to give information about
the figure of the earth and about the related subject of the formula for
gravity at its surfaced
OBSERVATIONS ON THE VARIATION OF LATITUDE
Ire the last decade of the ni~et.eenth century, observations for latitude
were undertaken by many of the observatories in all parts of the world
but the difficulty of making the results from these observatories closely
con~ara.ble soon became apparent because of imperfect knowledt,e of
the declinations of the stars used. It appeared desirable, therefore, to
use the same stars at all observatories. NYhen Taloott's method is used,
this is possible only when the observatories have practically the same
latitude. The conference of the International Geodetic Association held
at Berlin in 189.~ began the world of or~ani%in, the International Latitude
Service. The intention was to have special observatories well distributed
;
estimating the purely elastic yielding,. The material of the earth being what it
is, if the centrifugal force were removed, the equatorial bulge would be reduced
by about 3/10' of its present value.
* For details of the theory in the case of a yielding body, see Schweydar's paper
previously cited.
~ For an attempt of this sort, which yielded, however, only roughly approxi-
mate results, see W. D. Lambert, An invesl;~ga.tion of the latitude of Ukiah,
California, and of the motion of the pole. U. S. Coast and Geodetic Survey,
Special Publication No. 80, p. 59.
VARIATION OF LATITUDE
255
in longitude, situated in localities having, good conditions for observing,
and all on the same parallel. The stations finally chosen were as follows:
Longitude
Mizusawa, Japan ........................... 141° 08:' East
Carloforte, on a small island off Sardinia 8 09 East,
........ 77 12 West
........ 123 13 West
(3a~thersburg, Maryland ..............
Ukiah, California ....
All lie in latitude 39° 0S' North. Cincinnati observatory, being on tl~e
same parallel,ofleredto cooperate. Itslon~,itudeis84° 25' West. The
Russian General Stan offered to maintain a station at Charjui (otherwise
Tschardjui and various other transliterations of the Russian) in lon~,i-
tude 63° 29' East. Work was begun in the fall of 1899 or early in 190O
at all of these stations.
Three of the stations, Mizusawa, Carloforte and Ukiah, are still in
operation. During the World War they were operated under the auspices
of a Reduced Geodetic Association among, Neutral Factions, organized
to keep geodesy, as an essentially international science, alive during the
conflict, and more particularly for the purpose of operating, the inter-
national latitude stations. This organization continued until 1922, when
the latitude work betas taken over by the International Astronomica1
IJnion and by the Section of Geodesy of the International Geodetic and
Geophysical Union, both organized after the war.
Cincinnati Observatory felt obliged to withdraw its cooperation at the
end of :191~. The station at Gaithersbur~, was closed for reasons of
economy at the end of 1914.: The station at CharJui had to be sh;-ftccl
several miles because of a change in the course of the Amu Darya, a
large river near by; no word from the observatory came to the outside
world after 191~; it was then -found that it ha.] continued operations
until 1919. It is to be replacecl by an observatory some 3° to the west.
at Kitab near Samarkand, since for various reasons it has been found
impracticable to resume world at Charjui itself.
THE THEORETICAL CU:ELVE OF POLAR MOTION
As a preliminary to examining the results obtained by the Ixlterna.tiol~al
Latitude Service, it Play help to clari-iv our ideas of what to expect if we
examine a few theoretical curves obtained by the simple combination of
the annual motion with the fourteen-month Chandlerian motion.
:Fi.~,ure 2 ~ represents the position of the pole o-! rotation sullen the un-
disturbed pole of figure describes harmonically an annual path, repre
*Since this was smitten it has been announced that the Gaithersburg station
will be reopened in 1930 or 1931.
~ Figures 2, 3 and 4 are from NiVanach, die Chandlersche und die Newcombsche
Periode der Polbewegung, Berlin, 1919.
17
268
FIG~ 0F ~ ~
mcthods of observ~tion; to person~1 equ~tioD of obseTvers ~nd to ~cciJent~1
error; ~nd ho~ much to ~n~ re'1 cb~nge of l~titnJe it is very di~cult
to s~.
Sinco tbe Tntern~tion~1 L~t~uie Se~vice st~rte] tbere h~s heen one
c~se th~t Ior ~ nUmber oI ~e~s seeme] to ~epresent ~ ver~ slov But
continUous displ~cement oI ~e nortb pole tow~Tds \~th Amehr~.~
[o~ tbe ye~Ts 1900 to 1917; inclUsive, L~mBert IoUn] Ior tbe \oTtb
Fole ~ ~ver~ge motion of 0~.00~2 ~nnU~lly sQUtb~rd ~long tbe merid-
i~ of S1° WesL [or tbe perio] 1900-1923; inclusive, W~n~rb Tound
0~.0047 ~long tbe meTidi~n oI 42° WesL Tbe hrst 1) ~e~s oI ~is perio]
g~vo 0~.0025 ~ong tbe meriJi~n ~ 37° West; tbe l~at 13 ~e~rs of ~o
perio] g~ve O~.0040 ~long tbe meridi~n of 40° WesL Ibe tTen] is ~US
s~ll ~] Ji~cult to determi~e ~itb ~ccUr~cy; hut ~ssUreil~ re~l; tbere
no ce~t~D~ tb~t it ~ql continUe ~] its c~use ~ quite nnkno~n.
displ~cement of ~is so~t ~mounti~g to pe~b~ps one second of arc in 200
~e~s an] one mTnUte oT ~rc in 12~000 ~e~rs voUl] he of btUe Use in
expl~ining tbe l~st ~Dce of tbe icc in Eleistocene Ome, foT tbis occU~red
not ve~ m~ny times 12~000 ye~rs ~go; ~n] ~ displ~cement of the pole
of only ~ fev minUtes of ~rc ~oUl] b~ve only ~ negligible e~ect on tbe
clim~te. ~oreover; ve b~ve no ~ssur~nce tb~I even tbis Jov qU~s~secul~r
displacement existed in tbe >~st ~or tb~t it vill co~tinue in the IUtUrc.
OTBEG P8~0~A l~ 1B~ YARIATIO~ 0E [~IlIL[E
[eriods in tbe motion of tbe pole otber th~n tbose of one ~e~r ~n]
of ~pproxim~tely IoUrteen months, representIng respectively tbe Iorce]
~n] tbe free motions, b~9e been dedUre] from tbe observ~tions, bUt b~o
not yet IoU~d gener~1 ~ccept~nce, ~or b~s ~ny ~]equ~te expl~n~ITon heen
given ~s to ~by sUcb periods sboul] he pr~sent. Tbe ~pplic~tio~ of
Schnster~s criterion by ~dll~[ I sbovs th~t only the OnnU~l ~D] thP
fourteen-monib periods meet tbe iest.
Ibe phr~se ~ periods in ~e motio~ of tbe pole ~ ~s use] designeil~
iDstesd ~ {; periods i~ the v~ri~tio~ of l~titUie ~ tec~Use iU tbe l~tter
tbe~e ~ro periods otber tb~n those just mentioned; the~ ~re tbe ~e~iois
of tid~1 Torces th~t ~ect prTncip~ll~ the direction of the plUmt [net
Tb~ s~me [d~1 forces ~lso Jeform tbe e~rtb ~] cb~nge ~e poshion
* W. ~. [~mEcrt. An investig~tion of tbe l~thude of [ki~h, C~liforni~, ~nd of
tbe modon of tbe pole. C. S. Co~ ~d CeodeLc ~rv~ Speci~ P). ~o. 80, p.
36; ~lso, Ibe interpret~tioo of up~r~t cbunges in me~n l~Hude ~ ~c l~tern~-
hon~1 L~thude St~tions. Astron. J., vol. 34, no. 804: 103 (1922). B. W~n~cb. Fine
fortscbreitende [~geEnderung der Erd~sc. Z. Ge~bysik, 3:1~ (1927).
~ Leo W. Folluk. D~s Feriodogr~m dcr Folbe~egung. CeH~nds Beitr Gco-
physik. 16: 108 (1927).
FJ~IZO~ OF [~7~E
2G9
of the pole of Ague ty amounts by no Glenda in~precT~ble in comparison
with twos that bare been discussed. But the periods of the tiff forces
that Erect the position of die Pole of figure are so swat that the erect
on the position of the pole of rotation is negligible. Abe magnitudes
involve] are so smelt of the offer of 0~.001; abut ibex CAD he Drought
out only by Isis of great masses of otser`~ons. Ibe quantities
actually IOUP] in genera agree excellently, within Me inevitable Emits
of Uncertainty; with the values deduced from hi~1 tbeo~y And Irom the
assumed el~stio properties of the e~Tth.0 Some recent results ty Stetson
based on ~ genesis of photographic otservstions at the G~itbershn~g
lettuce station ales however, ~ exception. The b]~1 erect on the Jirec-
hou of the vertic~1 is mUrb larger tab theory predicts. No quantitatively
satisfactory explanation bus Been given of this anomalous result.
There is hotbed portion of the variation that is not due to the motion
of the pole. When the lager bus Been JeJUce] irom ~11 idle drip
and the v~i~ho~ of lupine Age ~ it bus Been subtype] from the
observed variation of latitudes some residual as repairs. It m~ he
called the non-pol~r variation of latitude. belt of the residual is, of
courses irregular and ~< he ascribe] to ~ccident~1 errors, gut to ~ certain
extent the ~on-pol~r v~ri~tTon of 1~titUie at one station resembles that
at ~notber, as iT ~11 stations increased or decreased their latitude simUl-
tenuously. Ibis non-pol~r v~liRtioU common to ~11 stations is called the
KimuT~ team $ Trod its discoverer. It is very smelt the semi-~mplitude
being of Me order of 0~.03 or 0~.04. Obvious explanations awe been
Once to account for iL ~ p~rti~1 explanation is certainly to he IoUnd
in tbc Ann allies of the st~rs; ~birb average about 0.5 magnitude
in the Tnternution~1 [~1itUie Schick and other reront observations. Out
these p~ll~xes cannot account Ior ~orc task on~-ToU~tb of the erect.
Ibc team might arise Irom the Outbox of combining the observations;
] from the presence of ~ possible gaily period in the Mention. StM1
~ _ _ ~ _ ~ ~ ~ _ _ , ~ .
.. . .. , . . .. ~ . . . . .. . ..
~notber expl~n~tlon 1lcs in the presence of ~D~1 nits Dean caUea ~ loom
rcTr~ction.~ Ibe erect of detraction depends amidst Molly upon the
condition of the ~tmospbcre i~edi~tcl~ outside of the telescope. In
recent years astronomers bane ~ecognizc] bow important it is to maintain
* Sce lo. PrzbylloL. Obor dig ids dcr LotEcweguDg. As~on. ~ucbrichten~
vol. 218, no. S214: 8S (1923).
T. Ibid Toni Hi. ~tsny~m~ Cb~nge of plumb line referred to the axis of the
Crib as found from the result of the Intern~tioDu1 Obtrude Observations.
Memoirs, College of Science and Engineering, Eyoto Imperial [niv~ vol. 4, no. 1,
1912.
~ B. T. dawn. On Me v~ho~ of lade gab the moods position. 8c
eucc, 60: 171 (1929).
~ H. Figure. Astro~omiscbe \~cbrichten, vol. 138, p 233.
270
FI CURE OF THE EAR TH
the conditions, especially with respect to temperature and ventilation,
exactly the same to the north and to the south of every meridian instru-
ment. This cannot be done so well in a small building as in a large.
Yamamoto,* of I(yoto, has recently tried the experiment of putting a
zenith telescope into a very long observing room, no less than twenty
meters in the north and south direction. Under these conditions he
observed for two years, and reduced his observations in the ordinary
way, with the result that little of the Kimura effect appears to remain.
GEOPHYSICAL SIG-NIFICANCE OF TEE VARIATION OF LATITUDE
An interesting, conclusion may be deduced from the lengthening of
the ten-month Eulerian period into the fourteen-month Chandleria~
period. This lengthening, as was suggested by Newcomb, is due to the
yielding of the earth, a yielding representing, the combined effect of the
flow of ocean waters to adapt their level to the new position of the pole
and of the yielding, elastic or conceivably plastic, of the remainder of
the earth. Newcomb, on the basis of data then available, estimated the
average rigidity of the earth as somewhat greater than that of steel. This
estimate appeared to be supported by early determinations of the rigidity
from long-period ocean tides and by the very accurate observations of
earth tides made by Michelson and Gale, and since the phrase " about
as rigid as steel " is rather striking and sticks easily in the memory, it
has gained a somewhat undeserved currency. Later determinations and
more careful calculation malice the rigidity about twice that of steel or
even more.l The different portions of the earth have, of course, different
moduli of rigidity and the estimate just given represents merely an average
value, an average, moreover, that is valid only under certain simplifying,
assumptions, for in problems like the present one the distribution of den-
sity within the earth is an important element. Nevertheless the statement
that the earth is on the average about twice as rigid as steel is as
satisfactory a brief statement as can be made at present.
What the latitude observations really determine is not a. modulus of
rigidity of the whole earth or of any part of it, but merely an abstract
number 7~, which may be explained as follows:
The disturbin,, forces due to the non-coincidence of the axis of rotation
with the axis of figure give rise to forces that have a potential Thy of the
form
9..O
IV`, = ~ S.,.
a
* Memoirs of the College of Science. Kyoto Imperial University, Ser. A vol. VI.
no. 7 (1922-23).
~ See W. Schweydar. Lotschwankung und Deformation der Erde durch Flut-
krafte. Zentralbureau der Internationalen Erdmessung. Berlin, 1921.
.`
VARIATION OF LATITUDE
271
where r is the distance of the point in question, a is the radius of the
earth treated as a sphere and S. is a surface spherical harmonic of the
second degree; in the present problem the harmonic is of special type
but the idea behind the definition of 7c allows any surface spherical
harmonic of degree two to be used. On account of the mobility of the
earth there is a shifting of matter produced by the force in question
and this shifting produces an additional potential of the same type, that
is, of the form AWN. The potential of the total disturbing force is now
(~+k)W~. Thus k is defined. To take a fairly familiar illustration, if
we had a rigid sphere having the same mean radius a, as the earth, and
the same mean density and rotating at the same rate, a, the flatt.enin~
of an outer level surface would be
Ma
f _ - _
' 2q ol8 '
approximately, where g is the acceleration of gravity at the surface. But
if the globe were homogeneous and plastic, the equatorial protuberance
would increase by its own scl:~-attraction until ec.~uilibrium was obtained
with a flattening f.' given by
r ~
r ~ ~-a
STY= 4-.
Since the harmonic terms are proportional to the flattenin~,s produced
by them, we have
(~+7~)W' f _
ti2 it ~
or
1~= 2 .
The ellect.ive value of: 1~ for the variation of latitude is -Iou~cl iron the
formula
1, _ (1 - ',,° ~ ~ i -1 )
Allele TO and T are respectively the ten-month Eulerian and the iour-
teen-month Chandlerian periods, f is the flattening of the earth and m
the ratio of the centrifugal force of rotation at the equator to gravity
there. As the best available values let us take To=303.3 and T=432.5
mean solar days, f=~/297 and m=~/289. The result is
k=0.276.
This value represents the combined effect of the elastic yielclin~, of the
solid earth and the mobility of the ocean waters. The 1~ for the solid
earth is perhaps equal to 0.22 or 0.23. It is Irom a value of it corrected
18
for the erect of the motILty of the ocean maters tat the modulus of
rigidity of the ebb shoal] he inferred. Obviously some Iurtber ~ssu~p-
tiona are required. Tbe simplest Irom the m~tbe~tic~1 point of Stem is
abut the Crib is homogeneous and incompressible. To get Twos this
assumption ~ value of ~ like 0.22 or 0.2) requires ~ rigidity about twine
tat of steel.^ Comp~essitility in this instance bus compel little
erects tbo~gb it goes bane ~ more appreciable erect on some of the
pbenomeD~ connected with ebb tides, under ~bicb cl~ssiEc~tion some
aspects of the Satiation of lie mitt Be include].
Ibe value of ~ gives us simply one iispos~tle parameter fig the
many that are required Iull~ to specify the distribution of density and
elastic properties within the glib. Abe compTexit~ of the teres~i~1
structure non most froze] t~ ge~b~sic~t~I has prevented ~ to
test of tab compatibility of their assumed structure vita the observed
value of a, But c~TcUl~tions made under simplifying ~ssnmpti~ns do not
surest tat there is likely to De any serious iiscorisnce.l
lo estimating the erect of the motiLt~ of the ocean maters or the
value of ~ it is no to assume Abut the Mater follows the static 1~;
Bat is; adjusts itself so that its sU~ce-~ from the erect of Cress
fives; etc. conforms to ~ level surface of the instantaneous geld of
force. Ibe change in level DUO to the displacement of the pole is so
smelt Being of the order of magnitude of one centimeter, abut c~ref
analysis of mean se-level is required. Ibe costlier investigators IoUn]
that within the limits of otseTv~tion~1 error mean se~-level did aura
with the shifting of the pole in conformity with st~tic~1 1~; tUt in ~
more recent and extended Tnvesti~tion P~zL>llok ~ awls the contrary
to he true Hi least Ior places along the Forth and Baltic Seas.
~ = ~10 ~
fibers ~ is the modulus of rigidity, ~ the radius, p the mcun demise of the
e~rtb, and ~ the ~cccI~ution of gravity.
ITbis populates ~ centru1 core braving vc~ low rigidity, p~rb~ps having no
Hasidic at ate, any more tb~n ~ liquid bus, but of ~ big dcgrec of incomprc~-
bUity. The radius of this core is more than bolt the radius of To earth. Outside
of this liquid or nearly liquid core comas Solid matter decreasing in density and
rigidity as the surf~cc is ~ppro~bed. There are disoontiDuides in this outer por-
don; their numbs and depth are variously given by se~mologi~s, to Boom this
picture of the interior of the eurtb is mainly due.
~ Harold Jerseys. The rigidity of the earthy contra core. To. Notices Boy.
AstroD. Coca Ceophys. 8upplcmcDt, 1: 371 (1026).
~ E. Przbyllok, fiber dig sogenunDte PolOut in dear Ost- and ~ordsee. )
VARIATION OF LATITUDE
273
It was once assumed without question that the statical law would apply
to oceanic tides having periods of a iortnit,ht or a month, although it
had been known that tidal oscillations of quite another type were theo-
retica.lly possible on a rotating globe. The question is one of length of
period and the amount of internal friction. If the period is long enough
and the friction great enough, the tidal oscillation will be of the statical
type, though with perhaps an appreciable lag in phase. It is now realized
that earlier estimates of friction were much too low and that a virtual
coefficient of eddy viscosity of turbulence should be used instead of the
much smaller coefficient of viscosity determined by ordinary laboratory
methods. NVhen the coefficient of turbulence-a very variable quantity
according, to circumstances--is used, the forthrightly and monthly tides
are probably not far from the dividing, line that separates tides of the
obvious statical type from tides of the other type. The fourteen-month
Chandlerian period should apparently be long enough for the tide to be
truly statical in the open ocean, so that Przbyllok's. results are somewhat
puzzling. The explanation may lie partly in the enclosed character of
the Baltic, which is almost an inland lake, for the levels of tide stations
on the North Sea conformed more nearly to the statical law, partly in
the smallness of the quantities dealt with, and partly in the fact that
changes of sea-level, whether due to meteorological or climatic causes
or to small hypothetical changes in the Pure of the earth, are themselves
causes in some degree of the variation of latitude. The question has many
aspects and needs further data and further study.
If we could study the decay of the amplitude of the Chandlerian motion
of the pole, we could then learn something about the viscosity of the
earth as a wll,ole, but as has been noted, the amplitude and phase of this
motion is subject to so many irregular perturbations that little can be
determined. Apparently, however, the coefficient of viscosity is at least
of the order of 10'° c.~,.s. 1lnits. If we write this coefficient as equal to
six T where it is the modulus of rigidity of the earth and T the so-called
" time of relaxation," and put it equal to 2 X 104~ we kind with a value of
the viscosity equal to 10~° that T = ~ x 10' seconds, or about a. year
and a half. The value of T. and with it the coefficient of viscosity, may
spell, however, be considered larger,* for la.r,e values of the amplitude,
when once established, seem fairly persistent.
Observations of the variation of latitude may be used for both geological
and astronomical purposes. In connection with suspected crustal move-
ments latitude observations would be especially useful if the movements
~ See three papers on The viscosity of the earth, by Harold Jeffreys. Mo.
Notices Roy. Astron. Soc., vol. 75: 648 (1915); vol. 76 (19~15), p. 85; Sol. 77
(1917), p. 449. See also, reference to Dyson on p. 204.
274
FI~ 0F r~ ~
were on ~ continent~I scales as tbey would he on the Wagerer b~potbesis.
Latitude observations would he less use1U1 for e~rthqU~Le displacements
or for the slowly accumulating strains that Ace suppose] to lead up to
sol e~rthqu~Le, because totb the slog strain ~] the sudden rebound
Pro relatively small and attain observable size such as ~ few meters-
only in [mitc] regions Scar lines or center of seismic activity. [or
determining carte movements of this sort, latitude otser~tions cannot
compete in ~ccUT~cy with good geodetic surveys of the region in question
repeated at suitable intervals provided gusts abut the survey can Start
from ~ line far enough Achy from the suspected region to he considered
as stable.
ASTRONOMICAL SIO\IFIC~CE 0E THE YARIATIO~ HE [~IIICDE
Obviously ~ stuffy of the variation of latitude is esse~ti~1 to the prcp~-
r~tio~ of ~ fund~ment~1 star c~t~logue. Speci~1 work on the subject like
abut done by the Intern~tion~1 Latitude Service, representing the co~-
hine] result from Bang otserv~tories, gill determine, as well as May be,
that Boat of the variation that is due to ~ motion of the pQlc, hut ebb
observatory ~y bade As own peculi~rides represented By apparent
changes of latitude; Both irregular and systematic over and above that
due to polar motion. Probably most of these apparent cb~nges are due
to refraction; hut the methods adopted in the study of the variation of
latitude are also well adapted to ~ study of the subject of retraction, which
is related to it not to say entangled with it.
Ibe Cb~nileri~n motion presents Bang unsolved problems the eluci-
d~tion of which would brow light on various questions in astronomy and
geopbysics. We should, for instances like to know more About the seom-
ingly irTe~l~r pcrturb~tions of the Cb~ndleri~n motion and to learn
what connection they May bade Tab cages in the Tic of rotation of
We earth, changes in se~-lc~el in various parts of the glote, canes in
the e~ttb~s magnetism; or with terrestri~1 cb~nges of And sort.
Iben; too, we sbonld like to surmount the di~cUlLes due to these
irregular perturbations in order to determine accurately the length of
the Cb~ndlori~n period in order to Stain an accurate value of the
qU~nti~ a. To use this qU~nti~ in connection with ~ study of the el~shc
properties of the earth, it is necessary to correct it for the Mobius of
the ocean waters, and here again ~ Study of se~-levels is needed. When
goad value of ~ bus hecn Optic] ~ gal Pectin to [c seen bow Ecu
it can tic Ottcd into the picture of the e~rtb~s interior Irma By seismolo-
gists and other geopb~sicists. Ibc calculation of the clitic yielding of
globe composed of conccntric spbcric~1 shells of different densities and
cystic ~ro~ertics r~i~l~ incises in ~n~l~tic~1 complexity an] in the
FJ~\ 0F [~7~
amount of sheer ~ritb~etic~1 work needed as the Canter of sbebs iD
cre~aes, being exceedingly laborious even Ior ~ small nutted, but no new
m~tbemstic~1 principles Ace needed.
Egging it voUl] Lc extremely interesting to determine ~betber the
u~0istUrte] Cb~nileri~ motion is circular or perceptibly elliptical, and
iU the latter case to determine the direction of the ages of the elapse
and their ratio. Such ~ dete~miD~tio~ Could throw light on the figure
of the es~tb and on the question of the presence or absence of ~ longitude
term in the Gavin IOT~U1~/' This metro] is probably not susceptible
of the ultimate ~ccU~c~-obt~in~tlo from ~ gravity survey covering the
Bedtime glote; including the oce~lls; gut it would be interesting to compare
the longitude team in the gravity formula ott~iDed Irom ~ Study of the
Cb~dleri~n motion with that obtained Iron ~ Gavin survey, because
the data and methods Ace so entirely different.
Finally; the qUcstioI1 of periods in the 1~tituie variation, other than
the ~n~, tab Cb~ndleri~U; ~] the known b]~1 periods, id of courses
got close]; ~ltbougl1 thee seems to be as get no adequate >root of the
existence ~ such peTiD]s. ~iSCUSSi~S ~ longs series of observes
needed, snd if the existence of ODE [OliO] iS indicated; the problem
n~tur~llv arises of Lading an adequate expl~n~tioG.
One of the most difficult problems of modern astronomy is to determine
] to eliminate systematic errors in star places. It is easy to see now tat
insufficient prcc~utio~s were t~kcn fly earlier observers with tab meridian
circle ~] similar Tents; as ~ result it is extremely ii~CUlt to disen-
t~ngle certain probable cosmic eJects from mere systematic error. It is
curious and interesting abut observations for Trio of 1~thU]e may be
use] to ego some of these systematic errors and are tbere10le capable
of tbro~ing some light on problems connected vita the distribution of stars
in our galaxy and with their motions. II we make the assumption tat
ale 1~titUJo of ~y station supers no secular cb~nges c~cc~t those that
Mov from Sac polar motion; thcD ~ long-continue] series of 1~titUic otscr-
v~tions will give us Sac pour motions of the stars obscr~cd, OD] with
high dcg~cc of precision. A single l~tituic pair can tic observe] with
pro)~tlc error of 0~.1 or less, OT, in over woodsy the mean declin~tiou
of the two Stalls COD tC mc~sUlC] vita abut precision. Comp~ti~cl~ flaw
obscrv~tious c~teDiing over comparatively few Seals will tberctorc Field
the Scan of the proper MOTORS of these stars with ~ motile error of
0~.001. We bloc belle ~ metbo] abode independent of the meridian circle
*For an attempt to determine the figure of thc crab in this any, sea W. A.
[smart, An investigation of the Etude of climb, Chic, and of the motion of
the pole. P. S. Coast and Geodetic Survey, Special Publication To. 80, pa 39.
270
] ~Ucb gore precise tb~n id In determining the proper motions of
stars. Ibese Stats ale necessarily few in Dumber but are still sufficient
to test the meriii~-circle proper motions. This ~etbo] bus already Been
use] to calibrate the proper motions of stars Bose Jeclin~tioDs corre-
spon] to the latitudes of observatories in the north L~per~te zonk that
L, from about + 30° to + 60°. It is bigly desirable to extend such work;
especially to soutbeln latitudes. A Beginning in this direction bus Are
beer made at JoL~nesburg, South Africa, where, twenty bears ~go, ~
series of latitude ~seTv~tioDs was Charlie] out Tor sevens Create. At
B. SchlesiDger~ suggestion certain of the same stars mere re-observe]
in 1920, and these observations sboul] Field bighly accurate proper mo-
tions for these stars. Schlesinger bus also suggested that ordinary l~titnie
observations such as brave been carried OUt for example, iD CoDDOCtIoD
with the geodetic surveys of the Baited States RD] magi, mi~bt he re
_ ~ ~ ~
petted with the oxpect~tioU of delving Irom them much v~lu~hle infor-
m~tio~ as to the proper motions of the stars concerned. The early
observations are somewb~t lass accurate tb~n abut bus recently heen
attained, hut are stM1 accurate e~oUgb to answer admirably for this
important purpose. It may be Ant the assumption of freedom from
variations that ale Dot due to polar motion ~i] turn out to he UnjUstiCed;
gut this would appear from the results themselves provided abut Jeter-
mi~tion~ of this kiD] mere forthcoming from ~ sufficient Dumber of
st~tiolls Dell distributed in 1~IitUie and longitude.
References to ~ number oT the more important pagers on the variation
of l~ti~Jo bare Been given in the Ioot-notes. Ibe papers tbemsclves
will supply IUrtber reIereDces. Ibe most convenient soUTte oI inform~-
hon Stout the otserv~tioDs of the InterD~tioD~1 Latitude Service is the
ResUlt~te des Intern~tion~le~ Freitendienstes (Zentr~lbure~ der Inter-
n~tion~len ErdmessUng), the East volume (BerED, 1903) By ~brecht;
the second by AlbrerEt ~D] W~cb, the concluding volume (~^ Y.
Berlin; 1916) by W~n~cb ~lone. Ibis gives the definitive results for the
dears 1900-1911, inclusive. From the beginning of 1912 to the autumn
of 1922 the only readily Idle information is given in pagers by
W~n~cb in the ~st~ooomiscbe ~brichten, generally pro~ision~1 results
for ~ Tear or ~ period of bents. Ibe deLDitive redaction in the period
from 1912 to 1922 Bus Bade under W~D~cb~ supervisor ~] was nearly
complete] at the time of bis Je~tb. It is boded that it will he complete]
within ~ few months. ID the autumn of 1922 the Fork gas gut in charge
F]~70\ OF [~ZI~E
of ~ joint committee nylon t~ the l~ter~tion~1 Astrono~ic~1 Lotion ~d
ty Me Section of OeoJesy of the Inter~tioD~1 geodetic ~G] Geopb~sic~1
Anion ~itb Kimul~ ~ cab. Its reports of ~ fen gages act, gem
the title ~ [rQ`ision~I Besults of the MOTH of the Intern~tio~1 Latitude
Scrvice In the born gel 39° 8' for the period .... /, They are
pntl~he] in be Jose J~rn~1 of ~stro~o~ ant Geophysics or in
tho [roceedings of the (Ju>~I~se) I~peri~1 academy.
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