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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
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.
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.
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.
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.
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
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.
~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 -{ = (<P'17) £1{ or ~c: c=P'R: PR as stated by Newcomb. There are thus three different poles that appear in the theory of the variation of latitude: 1) the pole of the axis of' instantaneous rotation; this is the pole that is located by astronomical observations; in Figure ~ it is tl~e point R or Ret. 2) The point that would be the pole of figure if the pole of rotation should happen to coincide with it,. This pole may be called the undisturbed pole of figure or undis- -l-urbed pole of ine)~l-ia. The requirement that the pole of' rotation should coincide in imagination with the pole of figure is necessary in order to
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
206 FIGURE OF THE EARTH seated by the short straight line along the x-axis of the Internatiollal Latitucle Service, q. e., the meridian of Greenwich; the semi-amplitude is 0".1 or the total range 0".2. The pole of instantaneous rotation is assun~ecl to coincide at the origin of time with the undisturbed pole of figure, which, however, is then traveling, with its greatest velocity. The tiny circles with the numbers alon~sicle indicate the position of the pole of: rotation after the lapse of the corresponding, number of x-ears. The /0,~ Yet ~ ~ { ~ ~ ~J~ \ \ ~ ~ ~ ~ ,, x FIG. 2. Motion of the Pole Ideal case. i' tiny circles without attached numerals inclicate positions at intervals of 0.1 year. The whole number of elapsed years may be found by counting, forward or back along, the lull line, which gives the path of the pole of rotation. It is to be noted that the pole does not again come into the origin within the time shown, although it comes rather close to it. This close approach is clue to the near commerlsura.bility of the annual and Chanclerlian motions (: years being, approximately equal to 6 Chancller- ian periods). A cusp appears at about 6.o years; the complete curve is not shown beyond 8 years but a small portion is given around 13 years,
F]~70\ OF [~E 237 at whim time ~ cusp again appears. Ibe gr~JU~1 increase add decrease of the r~Jius-vector are Mite cb~r~teristic of the pa~ actually observed. If the undisturbed pole of inertia goes not start out in coincidence with the inst~nt~ncous pole of rotations me ~ get instead of CUSPS either open concavities or closed loops; according to the initi~1 condihons as sbov~ in Figure 3. Obese give the paths between 6 and ~ Teals from the origin of flame; the same inter in Ibid the CUSP DCCUPS in [ignre 2. Suppose the u~dis~hc] pole of Ogre to describe instead of the strut line in Figure 2 the small circle gun with ~ IU11 age in Figure 4. Oiven proper iniO~1 conditions; the instantaneous Pole of rotation may also describe ~ circle, the larger circle Baboon by the Jotted line. Ibe period of motion in this dotted circle is one bear and thee is no lte~te incre~sc in the r~Jius-vector so abut the Cb~ndl~ri~n motion drops ont of start This is ~ possible s~eci~1 cases ~ougb Dot ~ topics \ 7~ /~ ~ ~ ~ FIG. 3. Erect of initial conditions OD the pub of the Folc. 1 one. Tbe line with the tiny circles accompanied by numerals represents the same motion of the unJisturte] pole of [gure; gut ~ Ji~erent inid~1 position of the inst~nt~nrous pole of rotations ~ position in ~bicb both poles coincide at the origin of time. Here again ~ CUSP OCCURS DO-COP 6 One ~ beaks liters any Again we hnJ the ~ltern~to increase ~D] decrease of the r~]ius-vector. Ibe CUSPS may be eliminate] by cb~nging the iniG~1 . . . conultloDs. I8~ OBSERYE~ MOTION OF 1UF BOLE Tbe rectum paths of the Tub pole of rotation Icon 1890 to 1900, from 1900 to 1912 ~D] {10~ 1912 to lulls as JeJUce] by Web from Me otscrv~tions of the I~tc~D~tion~1 L8t~U]C Service are shown in Midges 3, 6 and 7, res~ccti~cly. These CUP7CS ~) BC analyze] into ~nu~1 RD] {ourtccu-montb components. (Ibere are Is of course oUtst~]iDg residuals.) Ibe Alcott of an analysis Ior the annu~1 component of the
238 FIGURE OF THE EAR TH years 1900-1911 inclusive is shown in Figure S. The lull line of the outer curve shows the exact ellipse resulting from the use of the annual terms only; the outer dotted line represents the result of including the semi- annual terms. The loan numerals correspond to twelfths of a year or approximately to elapsed months, the position XII representing, Janu- ary i, and ~ about February 1. The smaller ellipse inside represents the !// ~ 7. ~ \ \\ it FIG. 4. A special case of polar motion. a.l~.nua.l terms ill the motion of the undisturbed pole of figure; tile small dotted banana-shaped figure represents. the motion of this pole when the semi-annual terms are included. These terms are seen to have a n~cl~ larger effect on the undisturbed pole of figure than on the pole of rotation. This is because these terms that. affect flue pole of rotation are dil-terenti- atecl with respect to the time before they appear in the formula for the undisturbed pole of figure, and hence the influence of terms of short
VAI:?IATIOiV OF LATITUDE 2~9 period is greatly magnified. It is questionable whether these short- period terms should be taken very seriously. The results are even more irregular when use is made of the positions of the pole of rotation as directly determined and not smoothed out by the process of harmonic analysis. The numerical values of the derivatives are then exceedingly irregular and this irregularity shows up strongly 10m Urn 0 ~ ~ / i; it= ~? ~ '.\l! -0,: -0,2 ~OJIl1 Yew -lo"? +0"2 '0,"3 43447 6~ \~ 15411< ~p0 ~ -1 1 1 f 0,"3 fO,'2 -0"1 Am lam T ~ ~T ~ ; ~; lam ~0 _ 3 ~ ~_ Em \2 ~9~0 1; 19 _ ~98,0~ -_ 1 '1 1_ l - 0,"1 -0,"2 -0,"3 x Arc. 5.-Observed path of the Pole, 1890-1900. (Taken from NVanacl~.) O. Em 0m in the position of the undisturbed pole o:t figure as inferred directly from the observed positions of the pole of rotation. Some positions of the un- disturbed pole of figure determined in this way as found by NVanach are shown in Figure 9. It seems quite improbable that one year differs from another as much as this figure indicates. The annual sl~iftir~g of the pole of figure is due to seasonal causes and climatic changes, and one year should be pretty much like another, especially when the average effect for the entire globe is considered. Of course, there may be perturba
2 60 FI GURE OF THE EAR TH t.ions from non-seasonal non-climatic causes, and of these more will be said hereafter. Attempts to evaluate a priory the effects of seasonal causes in displacing the pole of' figure and thence to deduce the motion of the pole of' ro- tation have been made by Spitaler, Jerseys, Schweydar, and Rosenhead. ~ Seasonal variation of barometric pressure rather than unsymmetrical distribution of snowfall appears to be the dominant cause o:t' the shil'tin`~, loin em 0 5 In am I ~I T I ~I ~_ ~3 -0, 2 - 0,'1 +0,'1 '0,,"2 I/ ~ I~ 3 \9 it <~: . 6 ~1 _ 1 _ 1 _,1 ~ +0,"3 +0,"2 fO,"1 ~-~"1 -0,"2 -0;3 Flc. 6. Observed path of the Pole, 1900-1912. (Taken from TVanach.) I/, by, m 5rn Tom * R. Spitaler. Die periodischen Luftmassenverschiebungen und ihr Einfluss auf die Lageanderungen der Erdachse. Erg~inzungsheft No. 137 zu Petermanns Mitteil- ungen, 1901. Harold Jeffreys. Causes contributory to the annual variation of latitude. Mo. Notices Roy. Astron. Soc. 76: 499 (1916). W. Schweydar. Zur Erklarung der Bewegung des Rotationspols der Erde. Sitzb. Konig. Preuss. Aloud. Miss., 1919, p. 357. L. Rosenhead. The annual variation of latitude. Mo. Notices Roy. Astron. Soc. Geophys. Supplement, II: 140 (1929).
VARIATION OF LATITUDE 261 of the pole of inertia, and other causes that might be thought unimportant are shown to be not wholly negligible. The agreement. of the polar motion thus dedu.cecl ~ priory with the actually- observed notion is as good as could be expected in view of the uncertainty of the cla.t.a. Thus far we have discussed only the annual component of the polar emotion. There remains the fourteen-month Chandlerian component, the combination of which with the annual component gives to the radius Section of the pole its alternative increase and decrease as in Figures --0,3 -0,"2 ,'! 1 +0,'1 fO,'2 f 0,"3 10m Am 1 ~ 1 ,7' '14 0 '\\\\~4 +0,"3 +0,'2 ~ . . . . . . . . ~ ~- . =~ B ~ 12,0 _< - 1 1 t 7_ , -0,^! -0,'2 ~0, 3 FIG. 7. Observed path of the Pole, 19il2-1918. (Taken from Wanach.) Em . . . O - - sm Mm 9 to 1:1. To ~,ive an idea of the n~agnitucle and the variability of the fourteen-month eon~ponent we write it in the harmonic form. x=cc cOS (`ct-a) ll-~ cos (jot-3') The coordinates are referred to the axes of the International La.titucle Service for the north pole as already explained. The annual motion of the pole of rotation is la.r~e in comparison with that of the undisturbed pole of figure because a year is not very clifferent from fourteen months, the natural free periocl; or because, in short,
262 FI CURE OF THE EAR TH the motion of the pole of rotation is ma~,nified by q~e.so~ccnce. Short-period displacements of the pole of figure, such as those due to tidal forces, are aln~ost without effect ore the pole of rotation whereas in very slow TV - ~ v! /' \ / Irk ~ // =~.~Q · 11 ~ \ x~ ~-~ ~ v . ~ 'my \ me, - x , , ~ LAY m~7~rfo.o324sJ /foof 0.00996 /see ~ 30.82 meters. /sec::; - /C/~/O feat. At\' FIC. 8. Annual component of the Polar motion, 1900-1911. it\ /' I,' / ~ I 1' l 1 1, I Ail''''' id= displacements of the pole of figure the pole of rotation shifts almost part passe. Years covered lay observations (inclusive) <t a b ~ 900~05 0'.'131 32.°7 0''141 310°6 906-11 0 206 23.3 0.217 293 4 912-17 0.179 52 8 0.205 312.0
VARIATION OF LATITUDE 263 The time -l is reckoned from the year 1900.00 as origin, and if the unit be the Julian year K hats the value 360 ° x 3 32~-5 = 304° 0 The fact that a and b are nearly equal a.ncl the difference ,8-.a is nearly 270°shows that the curve representin.,the 14-rnont7~compo?~ent~sr~earty . 7 c~ron,la,r-. No analyses of recent years are available, but Kimura. reports the amplitucle of the fourteen-month component to be small during the years 1924-27. These results are averages over six-year periods. Wanach * fit, Amp ~ (~: x x 1 905 1906 f , ~N : , in'" , ~,:~,~ ~3; f`, )1~? 1~2 °- 1~2 ~ X X l ~ W Fig. 9.-Path of the undisturbed Pole of figure. (Tal;en from Wanach.) endeavors to determine instantaneous values of the amplitude and phase. The results show even greater variability than those in the table above, because of the ejects of observational error in the necessarily short series of observations used. DISCUSSION In spite of all these years of worl: our knowledge of the theory has not advanced as much as knight have been expected since the days of Chandler and Newcomb. The length of the Euleria.n period of ten months, pro- longed into the Chandlerian period of fourteen months by the yielding * B. Wanach. Die Chandlersche und die Newcombsche Periode der Polbewe- gung. Zentralbureau der International Erdmessung. Pub. No. 34: 23 (1919).
264 FIGURE OF THE EARTH of the earth, seethed to be sli¢,htl~' variable. Chandler tried to' deduce purely e~pirica.1 laws for the variation in amplitude and phase of the annual and fourteen-month components, but it is now recognized that the phenomenon is subject to unpredictable irregularities. Even apart from the inevitable but not :-et quantitatively determineLl damping, out due to friction, the I'ourteen-month motion is variable in phase and also in an~plitude because something,, or ~ variety of' thins, disturbs it, just as the motion of' a pendulums swint,in~ on a ship at sea is subject to irregular disturbances by the shoals to which the vessel is subjected through the action of' wind and wave. The a.nnua.1 motion is variable because in the first place the cause is seasona1~.~1 meteorological ancl each year differs somewhat front every other, and in tl~e second because the irregular perturbations of' the l'ourteen-mouthly motion affect the annual motion also.' The causes of these perturbations, other than those that Plight be termed met.eorolo`~ical, are something, of' a mystery. They Alight be due to crustal movements of' sufficient ma<,nitucle, but the magnitude would have to be great so great that catastrophic earthquakes would pre- sumably result. Studies have been made attemptin', to correlate the occurrence of earthquakes, even comparatively minor ones, with the vari- ation of latitude,! and the su~,3estion leas been made that the stress due to the variation of latitude acts like the trigger of a `,un in which the application of a small force causes the release of much greater forces pent up in the crust, or in the charge of the gun. In the case of crustal stresses the " trigger " effects of this sort simply change somewhat the time of' occurrence of' an earthquake that would have occurred at about flee same time in ally case, even snort. from t.hn t.rio.o,~r affect Rare +1~P ~. .. . . . 1 ~ ~ ~ v_ ~ ~ t) ~ ~ ~ ~ ~ u - ~ ~ v ~'1 ~ ~ stresses due to the var:lat~on of latitude are relatively so minute Car that they would apparently count for little even as trigger erects. Perhaps the correlations, to wl~a.tever extent it. may exist, between earth- quakes and the variations of' latitude may be due to the fact that both * See F. W. Dyson. The variation of latitude. Bull. Calcutta Math. Soc., 20: 23 (192S); also The Observatory, 52: 79 (1929). The work of G. U. Yule, therein re- ferred to, is found in Trans. Roy. Soc. London, 226 A: 267 (1926-27), entitled On a method of investigating periodicities in disturbed series, with special reference to Wolfers sunspot numbers. ~ Larmor and Hills. The irre~,ul;~r movements of the Earth's axis of rotation. Mo. Notices Roy. Astron. Soc., 67:22 (1906); 75:518 (1915). V. Conrad. Zur Frage der Erdbeben Haufigkeit und Polbewegung. Gerlands Beitr. Geophysik, 18: 247 (1927). ~ by. D. Lambert. The variation of latitude and the fluctuations in the motion o~f the noon. J. Washington Acad. Sci. 17: 134 (1927). Also, The variation of latitude, tides and earthquakes. Proc. Third Pan-Pacific Science Congress, Tokyo 1926, p. 1517.
VARIATION OF LATITUDE 26)0 are due to some cause, not yet understood, working within the eartl~. There is now fairly clear evidence that there are irregularities in the rate of rotation of the earth * and the only plausible expla.nation is that the earth's moment of Inertia about It's axis of rotation is chan~,in~, since it' the angular velocity of rotation is chan,~,ed, the moment off inertia, must change in the opposite direction in order that the an¢,ula.r momentum mar remain constant This change in the monument of inertia means ail effective shrinking, or swelling, of the earth. Brown '' does not su=,~,est the cause of such a phenomenon. King ~ su~,ests. that it may be due to n~a~neto-striction, Chicle is the change in the dimensions of a body due to changes in its magnetic condition. He connects the changes in size (or moment of inertia) with observed changes in the terrestrial magnetic element and finds that there seems. to be a correlation between the two pl~enon~ena and that the n~.¢,neto-striction involved is about the right order of ma',nitude. For changing, the moment of inertia and with it the rate of rotation, all that is required is a chancre symmetrical about the axis; but in view of the lack of symmetry of the surface features of the earth it, seems likely that any change of this nature would be to a certain extent un- symmetrical and thus by displacing the pole of figure would bring about a variation of latitude. There is some evidence that irregularities in the · c~ variation of latitude are correlated with ~rre¢,ula,r~t~es in the rotation of the earth. But for both kinds of irregularities there is no general a~,ree- ment as to their cause, the quantities dealt with are so small as easily to' tee obscured by errors of observation, and accurate data for proving or dis- provin3 suggested causes are in general lacking. SECULAR ~IOTION OF THE POLE It would be perhaps a convenient explanation of various phenomena of' geological history if it could be assumed that the axis of the earth had in past time occupied a different position in the body of the earth. There are coal cleposit,s in Greenland, implying that there once existed a lux~- riant vegetation where now only stunted willows and birches can grow, even in the most favored locality, and there are many other evidences that the climate of a given place hats not been uniform. The great ice sheets * E. W. Brown. The evidence for changes in the rate of rotation of the earth and the geophysical consequences, with a summary and discussion of the deviations of the moon and sun from their gravitational orbits. Trans. Astron. Observatory Yale Univ., Cot. 3, pt. 6. by. de Sitter. On the secular accelerations and the fluctuations of the longitudes of the Moon, the Sun, Mercury and Venus. Bull. Astron. Institutes of the Nether- lands, vol. 4, no. 124: 21 (1927). YE. S. King. Rotation of the earth and magnetostriction. Nature, 123: 15 (1929).
266 FI GURE OF THE EAR TH that in recent geologic time coverecl Scandinavia and Northern Germany and in North America came down to the latitude of New York City ancl below, are evidences that chances in climate have been both great and widespread. 01 course, if the pole approaches any given region, thereby giving it presumably a colder climate, it recedes from a region in the same hemi- sphere but clifferin3 180° in lon~,it.ude, thereby placing, the latter region nearer the equator. Geologists are not agreed in so correlating, with respect to time the changes that have occurred in the climates of different regions of the earth as to leave a displacement of the pole the only acceptable explanation. For instance, current trend of geological belief is towards the idea that the advances and recessions of the Pleistocene ice sheet were nearly simultaneous in North America and in Eurasia. Nevertheless the hypothesis of a shifting pole is obviously too co~- venient to be overlool~ed. In the early days of geology, Buffon and others of the "catastrophic school" of t,eolo~,ists were advocates of such an explanation. They had successors, such as the advocates of the so-called " pendulation theory," a theory according, to which the pole of the earth vibrated like a pendulum through a definite large orbit on the surface of the earth, though the periods might be enormous in length, humanly speaking,. Astronomers in general remained somewhat sl~ept.ical, realizing that considerable shifts of mass would be needed to change the pole of figure by any considerable amount. Darwin put the matter in convenient and definite form.'7 His calculations were discouraging to the advocates of the theory of large and frequent wanderings of the poles, his maximum allowance of polar movement being 1° to 3° in any geological period, and the redistribution of mass required even for this relatively small displacement is almost too large for most geologists to accept. Attempts were made to diminish the force of Darwin's argument. Sir Willia.rn Thomson was invoked as authority in favor of large polar dis- placement.s; but the passage often quoted,! if examilled in connection with the context, makes it clear that the " ancient times " referred to were very ancient indeed, probably pre-geological, and that for those ancient times Thomson was willing to accept as probable the displacements of mass necessary to displace the pole by large amounts. Another line of argument was followed by Schiaparelli.+ Darwin's main calculation was made on the assumption of an absolutely unyielding, k G. H. Darwin. On the influence of geological changes on the earth's axis of rotation. Phil. Trans., pt. I, vol. 167: 271 (1877); or, Scientific Papers, vol. III, p. 1. ~ Sir William Thomson. Mathematical and Physical Papers, vol. III, p. 333. This is a composite of two of his papers and represents his revised views. :L G. Schiaparelli. De la rotation de la Terre sons ['influence des actions geo- logiques. St. Petersburg, 1889.
VARIATION OF LATITUDE 267 earth. :For this ideal case Schiaparelli obtains results similar to Darwin's, but he goes on to consider the other extreme case of a liquid earth, and the intermediate case of a plastic earth; and in both cases he falls into error. :For a liquid earth Schiaparelli bases his reasonin, on the Eulerian ten-month period, which has no meaning in such a case. For a plastic earth he mistakes the direction of the forces brought into play by the non-coincidence of the pole of rotation with the pole of figure. These forces tend to bring the latter into coincidence with the former, but Schiaparelli makes these forces tend to displace the pole of figure away from the pole of rotation, which he would leave trailing behind the pole of figure during the latter's wanderings. Darwin's calculations then still hold for the case of an unyielding, earth. Any departure of the matter composing, the earth from this ideal condition towards greater mobility would require greater not less dis- placements of matter to effect a given displacement of the axis of rotation in the body of the earth. There remains, however, a mistake in Darwin's paper for the case of a mobile earth which, rather surprisingly, seems to have escaped the notice of those who would like to assume large polar clisplacements.* Darwin hi.mse].f in summarizing, his conclusions did not accept the result he had obtained, but this is merely because he did not believe the earth during, geologic time to have been sufficiently plastic for his reasonin, to apply. The error does not come, as in Schiaparelli's case, from an attempt to treat a complex mathematical subject without using much formal mathemat.ies, but from an oversight in inteorn.tion clurin~, the ~na1 stages of the discussion.! completely to chance the nature of the conclusions. _ · ~ The effect of the error i s ~ ,, ~ There is but little adequate and direct evidence from astronomic al observations bearing on the question of a secular shi:ftin~, of the pole. Since the days when accurate astronomical observations began to be made there have been apparent changes in latitude at various observatories when redetermina.tions have been made; these chances are in general only a few tenths of a seconcl. I-Iow much is due to uncorrected perioclic mo~rements. of the pole, how null to differences in the tables of reirac- tion=,~ used, how much to Ides in star ca.talo¢,s used, to changes in Several such presentations of the case for large polar displacements might be cited, in none of which was this particular passage from Darwin quoted. ~ In the equations (not numbered) for x and y that follow equation (4) of § 5 of the paper referred to, a factor ,u ~ ~ is lacking in the denominators of certain terms. ~ Perhaps the growth of cities near established observatories has affected the refraction.
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. )<erof- fentUcLuDg des Frcussichcn Ceod~t~cheD IDst~uts. To. S0> 1919. IBM COIN CoDt~ns reference to cagier inve~igutioDs.
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. #
