ZOOLOGY: WEISS AND GARBER

Hancox, N. M., Biol. Rev., 24, 448-471 (1949).Hawn, C. v. Z., and Porter, K. R., J. Exp. Med., 86, 285-292 (1947).Lewis, F. T., Anat. Rec., 55, 323-341 (1933a).Lewis, F. T., Proc. Am. Acad. Arts & Sci., 68, 251-284 (1933b).Lewis, W. H., in Genetic Neurology (P. Weiss, ed.), University of Chicago Press, 1950,

pp. 53-65.Matzke, E. B., Am. J. Bot., 34, 182-195 (1947).Pitts, R. F., Biol. Bull., 64, 418-423 (1933).Porter, K. R., Claude, A., and Fullam, E. F., J. Exp. Med., 81, 233-246 (1946).Schmidt, W. J., Die Bausteine des Tierkorpers im polarisierten Licht, Bonn, 1924.Thompson, d'Arcy W., On Growth and Form, Cambridge University Press, 1942,

1116 pp.Weiss, P., Roux' Arch., 116, 438-554 (1929).Weiss, P., J. Exp. Zool., 68, 393-448 (1934).Weiss, P., Growth, 5, (suppl.) 163-203 (1941).Weiss, P., Anat. Rec., 88, 205-221 (1944a).Weiss, P., J. Neurosurg., 1, 400-450 (1944b).Weiss, P., J. Exp. Zool., 100, 353-386 (1945).Weiss, P., in Chemistry and Physiology of Growth (A. K. Parpart, ed.), Princeton Uni-

versity Press, 1949, pp. 135-186.Weiss, P., Quart Rev. Biol., 25, 177-198 (1950).Weiss, P., and Wang, H., Proc. Soc. Exp. Biol. & Med., 58, 273-275 (1945).Willmer, E. N., J. Exp. Biol., 10, 317-322 (1933).

ERRA TA

In my article, "A New Discussion of the Changes in the Earth's Rate ofRotation," in the January issue on page 9 in line 13 from the bottom 7 milesshould read 2.5 miles and in line 10 from the bottom 200 miles should read80 miles.

DIRK BROUWER

280 PROC. N. A. S.

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

Dow

nloa

ded

by g

uest

on

May

23,

202

1 D

ownl

oade

d by

gue

st o

n M

ay 2

3, 2

021

PROCEEDINGS /OF THE

NATIONAL ACADEMY OF SCIENCESVolume 38 January 15, 1952 Number I

A NEW DISCUSSION OF THE CHANGES IN THE EARTH'S RATEOF ROTATION

BY DIRK BROUWER

YALE UNIVERSITY OBSERVATORY

Read before the Academy, November 6, 1951

1. Three types of changes in the rotation of the earth are known to exist:(1) a gradual lengthening of the day; (2) irregular fluctuations in the rateof rotation, the accumulated effect of which has amounted to approxi-mately 430 sec. during the past three hundred years; (3) seasonal changesin the rate of rotation, causing the earth's rotation to be ahead by about0506 in November and behind by nearly the same amount in May. Theexistence of this type of change was first discovered by N. Stoykol in 1937,and has been confirmed by results obtained in the Greenwich Time Service.2

G. I. Taylor3 and Harold Jeffreys4 showed that tidal friction in shallowseas is adequate to account for the rate at which the length of the dayappears to have increased since the date of the most ancient astronomicalrecords available.W. H. Munk and R. L. Miller5 showed that seasonal fluctuations in at-

mospheric and oceanic circulation are of the required order of magnitudeto explain the observed changes with an annual period.The explanation of the irregular fluctuations has been more elusive.

W. de Sitter6 represented the observed fluctuation curve by a series ofstraight lines. This representation requires abrupt changes in the rate ofrotation of the earth. Sir Harold Spencer Jones showed that more accu-rate data than de Sitter had available did not confirm de Sitter's represen-tation in detail, and that this representation could only be considered arough approximation.

This is one of several aspects of the problem that remained in doubtuntil the two important contributions by Spencer Jones in 19327 and 1939.8In the former of these papers Newcomb's discussion of occultations of starsby the moon was revised. These results added significantly to our presentknowledge of the fluctuation curve. In the latter paper it was clearly es-tablished that the fluctuations in the mean longitudes of the sun, Mercury

ASTRONOMY: D. BROUWER

and Venus correspond to fluctuations in time identical with those corre-sponding to the fluctuations in the moon's mean longitude. It was alsoshown in this paper that if for the secular acceleration term in the moon'smean longitude the value obtained from a discussion of ancient observa-tions,4 + 5"22 T2, is adopted (T expressed in centuries), then the modernobservations of the sun and the inner planets require a term + 1"23 T2

s+30 |l1.

. ~~~~a+20 .. _ _ . _ _

410

-0

-zo 1 I-20

-30 -1~

-401450 70lo 1750 1S00 1650 190010

S

Il _10 1700 1790 SW NM 1900 N19S

FIGURE 1

Upper curve; fluctuation curve in the rotation of the earth; the dashed line representsthe parabolic solution. Lower curve: the dots represent the derivatives obtainedfrom a nine-point formula, the straight lines the derivatives of parabolic arcs fitted tothe fluctuation curve.

in the sun's mean longitude. In the mean longitudes of the planets thecoefficient equals that for the sun multiplied by the ratio of the mean mo-tions of the planets to that of the sun. These ratios are 4.152 for Mercury,1.626 for Venus. An exhaustive discussion of Mercury by G. M. Clemence9confirmed Spencer Jones's conclusion, both as to the ratio of the fluctu-

2 PROC. N. A. S.

ASTRONOMY: D. BROUWER

ations in seconds of arc in the mean longitude of Mercury to those in themoon and as to the coefficient of TP.

Since 1820 annual values of the fluctuation curve are available. Theaccuracy increases considerably about 1850, 1880 and again in 1923 whenE. W. Brown's new tables were introduced for the calculation of the lunarephemeris. 0

The dots in figure 1 (b) give the derivative of the fluctuation curve in theearth's rotation obtained by fitting by least squares solutions parabolic arcsthrough each nine consecutive annual values of the fluctuation and takingthe derivative for each mid-point. These derivatives can be approximateonly due to the rounding inherent in the nine-point formula. However,a straight line character of the derivative is suggested. By successive ap-

s

I mlv ec on *. dst. I

prxiaton paaoi arc wer the fttedto th flcuto1creoe

*%*..%.

a~ I I I I1 10 10 00 300

FIGURE 2

Cumulative effects of random deviations, normal distribution a = 0.15.

proximations parabolic arcs were then fitted to the fluctuation curve overintervals for which the derivative appeared to be nearly a straight line.The derivative of the parabolic sections curve is represented by the straightlines in the same figure. The fluctuation curve is remarkably well repre-sented by this series of parabolic arcs.

In the course of a further study of this problem I was led to examinewhether the apparent straight-line character of the derivative of the fluc-tuation curve could be accounted for by postulating random cumulativechanges in the length of the year. If F(k) represents the fluctuation curvein seconds of time for successive years, the differences should then be repre-sented as in the table below,

VOL. 38, 1952 3

ASTRONOM Y: D. BROUWER

F(O) AF(1) A + 5iF(2) A + 51 + 52F(3) A + 51 + 62 + 53F(4)

the 6's having a mean value zero and standard deviation a to be obtainedfrom a discussion of the observational data.

Dr. A. J. J. van Woerkom and I examined extensive samples of accumu-lations of random numbers, a selection of which is given in figure 2. Thesimilarity with the derivative of the fluctuation curve is sufficiently strikingto suggest that the hypothesis may be compatible with the fluctuation phe-nomenon.

2. If the proposed interpretation of the fluctuations in the earth's rateof rotation is correct, then the sum of the random increments 5 after n yearsshould have a root mean square value nl"'a. In the fluctuation itself,

F(n) = F(O) +nA+ (n- 1)1+ (n-2)62+ ... + an-1;the root mean square value of the double accumulation of the 5's is givenby

3 2 6

For large values of nal -- 3-'/2n"l' af

Thus F(n) - F(0) - nA may be expected to have an amplitude propor-tional to IT1'l", if T is the time in centuries counted from any epoch.A solution was made from the values of the fluctuations derived from the

observed motion of the moon during the last three centuries and from rec-ords of eclipses back to 720 B.C. For the data before 1681 I used the resid-uals furnished by Simon Newcomb,10 corrected to correspond to the sys-tem that was adopted by Spencer Jones7 in his latest discussions. Thiscorresponds to taking the moon's mean longitude from Brown's tables cor-rected by

ALa = +22!83 + 23!40T + 5!22P - 10!71 sin (140°T + 20?7),the last term of which eliminates the empirical term present in the lunartables. The correction to the sun's mean longitude from Newcomb'stables is

AL0 = + 5!20 + 5!43T + 1!23T2.

In these and all following expressions T is counted in centuries from 2000A.D.

4 PRCoc. N. A. S.

ASTRONOMY: D. BROUWER

Modem observations alone do not permit the evaluation of the T2 terms.If they are to be modified, the following changes must be made, as statedby Spencer Jones8:

in the moon's longitude, +5!22T2 -_ (+5!22 + s)T2;in the sun's longitude, +1!23T2 -* (+1!23 + .0748s)T2,

the factor 0.0748 being the ratio between the mean motions of the sun andthe moon. In order to agree with the modern observations the fluctuationin the moon's mean longitude must be changed correspondingly, B -0 B -

sT2.The solution was made by the method of least squares from the observa-

tion equations

x+ Ty+ T2z= F,

F being the fluctuation in seconds of time in the earth's rotation, relatedto the fluctuation B in the moon's mean longitude by

F = 1.821 B.

The choice of zero point of T, 55 years beyond the date of the latestequation used, is arbitrary. Choosing the zero date too near a date forwhich an equation is available is undesirable for obvious numerical rea-sons. The solution is to be made on the assumption that the mean errorof F increases with (- T)'a". Hence the equations were multiplied by thefactor (- T) - /2, and the right-hand members of the new equations weretreated as having equal mean errors. The coefficients and right-handmembers of the equations solved,

ax + by + cz = 1,a= (-T)/t, b= (-T)-1'/ c = (-T)+1 1= F(-T)are given in the table. The solution obtained is

x = -54S6 4 0.544 j *,y = -40.9 L 0.645,u*,z = -5.46 i 0.109,u*,

in which ju* is the mean error of the right-hand members I of the observa-tion equations. (Throughout this paper the mean error will be used toindicate the uncertainty of a numerical result.) The table gives the resid-uals v* of the I-equations as well as the residual fluctuations F*.The mean error obtained from the residuals v* is 1 618. While it is

permissible to take ju* equal to this value to indicate the accuracy withwhich the past record of the earth's rotation is represented, other aspectsof the problem require further discussion. The reason is that the randomprocess may produce a spurious quadratic term,

VOL. 38, 1952 5

ASTRONOMY: D. BROUWER

3. In order to solve this and several related problems that arose in thisinvestigation, Dr. van Woerkom and I constructed artificial fluctuationcurves with initial values F(O) = 0, A = 0 froth a sequence of randomS's with mean value zero and a normal distribution. Equations were thensolved with the same left-hand members as the equations in table 1, butwith the right-hand members taken from the artificial fluctuation curves

TABLE 1

EQUATIONS USED FOR THE PARABOLIC SOLUTION FROM FLUCTUATIONS IN THE EARTH'SROTATION

F a b c 1 7*

-661' 0.00719 -0.1929 5.176 -43 8 +16'0-2051 0.00861 -0.2049 4.877 -17. 7 +1.0-1472 0.00977 -0.2139 4.682 -14.4 +2.9-1339 0.01242 -0.2313 4.314 -16.6 -1.8-517 0.0256-297 0.0284-617 0.0310+42 0.136-5 0.143-53 0.146-22 0.149-38 0.155-27 0.161-23.1 0.176-7.1 0.202+3.5 0.222+11.3 0.236+15.8 0.249+20.4 0.270+25.5 0.295+27.5 0.318+26.2 0.335+23.5 0.358+20.8 0.388+15.9 0.432+9.9 0.472+6.7 0.518+5.4 0.573-1.0 0.637-15.4 0.715-22.2 0.811-28.1 0.929-25.6 1.080-19.2 1.276-21.9 1.540-29.5 1.908-36.6 2.452

-0.2944 3.386 -13.2-0.3047 3.269 -8.4-0.3143 3.187 -19.1

-0.5154 1.953 +5.7-0.5220 1.905 - 0.7-0.5271 1.903 -7.7-0.5290 1.878 -3.3-0.5378 1.866 -6.9-0.5442 1.839 -4.3

-0.5614 1.791 -4.1-0.5858 1.699 -1.4-0.6061 1.655 +0.8-0.6183 1.620 +2.7-0.6300 1.594 +3.9-0.6456 1.544 +5.5-0.6664 1.505 +5.0-0.6821 1.463 +8.7-0.6948 1.441 +8.8-0.7096 1.406 +8.4-0.7298 1.373 +8.1

-0.7560 1.323 +6.9-0.7788 1.285 +4.7-0.8029 1.244 +3.5-0.8308 1.205 +3.1-0.8600 1.161 -0.6-0.8938 1.117 -11.0-0.9326 1.072 -18.0-0.9754 1.024 -26.1-1.0260 0.975 -27.6-1.0846 0.922 -24.5-1.1550 0.866 -33.7-1.2402 0.806 -56.3-1.3486 0.742 -89.7

F*+2327'+202+370-100

-5.4 -192-1.5-12.9

+2.7-3.8-10.9-6.6-9.3-7.7-7.7-5.1-2.8-0.9+0.4+2.2+2.1+6.1+6.5+6.6+6.9+6.8+5.6+5.7+7.0+5.3-2.4-6.0-9.7-5.3+5.8+7.8+1.5-7.0

-37-401+22-25-73-42-58-47-42.1-24.1-11.9-2.9+2.7+9.1+16.2+20.1+20.0+19.0+18.2+16.0+12.2+11.3+12.4+8.5-3.2-7.3-10.3-4.9+4.6+5.1+0.8-2.9

for the same values of T in terms of 100 steps, only the sign of T being re-

versed. From solutions based on fifteen independent artificial fluctuationsequences it was found that the root mean square value of FT-'/2 is 5.3 4

3.1 times the root mean square value of the residuals after substitution ofthe quadratic solution. Thus the standard deviation of v* multiplied by5.3 gives the root mean sq^uare value of the accumulation after 100 years,

YEAR

-683-380-189+ 135850927986

162116351639164516531662168117101727173817471760.91774.11785.51792.61801.81811.9f825183518451855186518751885189519051915192519351945

T

-26.83-23.80-21 .89-18.65-11 .50-10.73-10.14-3.79-3.65-3.61-3.55-3.47-3.38-3.19-2.90-2.73-2.62-2.53-2.391-2.259-2.145-2.074-1 .982-1.881-1.750-1.650-1.550-1.450-1.350-1.250-1.150-1.050-0.950-0.850-0.750-0.650-0.550

6 PROC. N. A. S.

ASTRONOMY: D. BROUWER

IL = 365 4 21s. The standard deviation of the random change in thelength of the year that corresponds to this result is ao = 0s063 b 0s037.A second method of evaluating o- is from the derivatives for the last 129

years of the fluctuation curve in the earth's rotation. For 25 dates withan interval of 5 years the values of the derivative F, with unit of time oneyear, were taken from both the nine-point formula and from the straight-line representation. From these the derivatives F*, corrected for the quad-ratic solution, were obtained by 100F* = 100F - y - 2zT. A leastsquares solution from the 25 equations

-lOO2a*2_-100T

gave for the two solutions v* = 0!053 and 0s054, respectively. Again,this is a standard deviation reduced by the elimination of the quadraticsolution. Comparisons with the artificial fluctuation curves gave for thiscase the ratio a/a* = 2.2 i 1.2. The derivatives therefore lead to a stand-ard deviation of the random cumulative changes in the length of the year,= 0s12 i 0s06. Although the formal uncertainty of this determination

is greater than that obtained from the fluctuation curve, the two valuesmay be entitled to nearly equal weight. Hence a = 0s09 0s03 may bean acceptable compromise. A value of o- = 0!12 is, however, not ex-cluded. The appearance of the derivative curve favors this larger value.

4. The coefficient z of T2 obtained in the solution may be consideredto be made up of two parts: z = z, + Za, Zr being due to the tidal retarda-tion of the earth's rotation, za the spurious effect introduced by the ac-cumulation of the random process. Only the sum of the two parts is ob-tained from observation. It is possible, however, to give the root meansquare value o,, of ZaLet the earliest date used in the solution be denoted by - T, From

the representation

F(T,) = x + yTc + ZTe2+ F*(T,),it can be shown without difficulty that the root mean square value of z, de-termined from fluctuation sequences with F(0) = 0, A = 0, is almost ex-clusively determined by F(T,). Thus, approximately,

1000

the right-hand member of which is the root mean square value of the ac-cumulation in F after 100 TC steps, or

1000as = N/_3Te

a.

7VOL. 38, 1952

ASTRONOMY: D. BROUWER

With T, = 26.83, it follows that with

a= 0509, ur = i 1050;4-=0512, as = i 13s3.

The value of as was confirmed by evaluations of z from the fifteen artificialsequences. It was then also found that there is no significant relation be-tween the value of z obtained in the quadratic solution from a fluctuationsequence and the factor by which the root mean square value of FT-" isreduced by the substitution of the quadratic solution.The proper interpretation of the result for z is, therefore,

Zr = -5S46 o4s 0.109 ,*.

If as = 10oo, ,* = 658,zr = -5 46 41-0O,

the contribution due to u* being insignificant. The correction to the co-efficient of T2 in the moon's mean longitude becomes sr = -3!00 + 5"5.Further comments on this result will be found in section 7.

5. The large corrections found call for further examination. WhenNewcomb made his solution, he solved for five unknowns, three of whichcorrespond to x, y, z of the present discussion. The two additional un-knowns were to define the amplitude and phase of a sine curve with periodof 275 years. A provisional discussion had yielded this period as in bestagreement with the empirical representation of the fluctuation curve duringthe last three centuries. Solving for the five unknowns, Newcomb founds = -0!47. If his equations are solved with x, y, z as the only unknowns,the solution gives s = -3!30. Newcomb's data are not immediately com-parable with those used in the present discussion, and in the choice of theancient observational data his opinion differed from that of others who dis-cussed these records. The change of about -3' in the solution broughtabout by omitting the unknowns connected with the empirical term withperiod 275 years is, however, significant.Newcomb's solution was adopted by E. W. Brown with slight modifica-

tions in the empirical term and in x, y, z. In Fotheringham's evaluation ofthe coefficient of T2 in the moon's longitude from ancient observations, themoon's mean motion at 1800 or 1900 was taken to correspond to Brown'sformulae, again with minor modifications. The system in which the fluc-tuations B in the moon's mean longitude have been reckoned has thus re-sulted from an attempt to represent the last three centuries by a sine curve.

If the fluctuation is to be considered the result of random cumulativechanges in the length of the year, then the 275-year term has no real mean-ing, and a revised solution without the two additional unknowns to allowfor an adjustment of amplitude and phase of this spurious term is to be pre-

8 PROC. N. A. S.

ASTRONOMY: D. BROUWER

ferred. It is this modification rather than the new weighting of the, olderobservations that has produced the large changes in the solution.The solution obtained must be considered a provisional result. I used

the same selection of the older observations as had been used by Newcombbecause they were available in a suitable form. A new solution which Ihope to undertake in collaboration with Dr. E. W. Woolard of the U.S.Naval Observatory should include all of the available older observationsfrom the earliest records to the time of the invention of the telescope.A new reduction of these observations directly with Brown's tables for themoon and Newcomb's tables for the sun will be desirable. In previousdiscussions of the secular acceleration problem the emphasis has alwaysbeen on the better timed oldest observations available. The most cele-brated ancient eclipse was the one observed by Hipparchus in 128 B.C., towhich Fotheringham" refers as the critical eclipse of Hipparchus. Thetacit assumption was that the amplitude of the fluctuation curve could beconsidered to have remained the same through twenty or more centuriesas during the three centuries for which accurate, more or less continuousobservations exist. If the amplitude of the fluctuation curve must beconsidered to increase proportionately to (- T)"/2 from the present epoch,the principal uncertainty is that due to the accumulation by the randomprocess rather than the observational uncertainty. Hence the best deter-mination of the secular acceleration in the moon's motion is to be obtainedfrom a collection of reliable observations as well distributed in time as pos-sible rather than from the most ancient and best timed observations.

6. While changes in atmospheric and oceanic circulation appear to beadequate to explain the small seasonal changes in the earth's rotation, thiscause must be ruled out as a possible explanation of the cumulative randomchanges in the length of the year. The mass of the earth's atmosphere is8.6 X 10-7 times the mass of the earth. If one-half of the atmosphere, be-tween latitudes 30°N and 30°S acquired a drift velocity of 7 miles per houreastward, the length of the year would be decreased by 0.09 second. Inorder to account for changes in the length of the year as observed duringthe past century, drift velocities with a range of about 200 miles per hourwould be required. Over longer intervals of time the cumulative effectwould require even more prohibitive drift velocities.

It has long been recognized that the cause of the changes in the earth'srate of rotation that produce the fluctuation curve must be sought inchanges in the moment of inertia about the axis of rotation. If the slowgradual changes in the length of the day by tidal friction are ignored, thecondition is that from year to year

CO = constant,

C being the moment of inertia about the axis of rotation, Q the angular ve-

VOL. 38, 1952 9

ASTRONOMY: D. BROUWER

locity. of rotation. For a sequence of successive years the average valuesof C and Q will be

Co QoCo(1 + 'E) % (I - E1)Co(I + 'E + 62) QO(1 - El - E2)Co(I + + E2+ E3) ( - 61 - 62 - E3)

the E's being random increments with mean value zero and a standarddeviation 2.9 X 10`9 to account for a random cumulative change withstandard deviation 0.09 second in the length of the year. These 5's maywell be accumulations of smaller random changes with intervals muchsmaller than one year. The astronomical evidence does not throw furtherlight on this question.The result of the preceding discussion is, therefore, that the observed

irregular changes in the earth's rate of rotation may be the result of minutecumulative random changes in the moment of inertia about the axis of ro-tation. No large sudden changes of any sort need be postulated. Thismay, unfortunately, make it difficult to find a connection between geo-physical phenomena and the physical cause of the changes in the earth'srate of rotation.

7. The uncertainty in the coefficient of the T2 term in the moon's meanlongitude has been found to be much greater than had been obtainedformerly. Owing to the relation between the coefficient of the T2 term inthe sun and that in the moon as found by Spencer Jones from a discussionof the modern observations, this does not bring into serious questionwhether or not the earth's rotation is being slowed down by tidal friction.The provisional result Sr = -3"0 4 5'5 gives for the T2 terms due to tidalfriction and for prediction of the future:

in the moon's mean longitude, (+2'".2 4+ 5"5) T2;in the sun's mean longitude, (+1"01 +t 0'41)T2.

The secular acceleration in the sun's mean longitude being entirely dueto the secular decrease of the earth's rate of rotation gives for the changein the length of the day +0500135 + 0500055 per century.One of the principal theoretical difficulties in this connection has been

that of accounting for the ratio between the coefficients of the T2 terms inthe mean longitudes of sun and moon. Jeffreys12 gave the theoretical rela-tion

K- 3N + N,v K n

N + N1 ni'-N and -N1 being the retarding couples acting on the earth due to the

10 PROC. N. A. S.

ASTRONOMY: D. BROUWER

lunar and solar tides, respectively, n and n1 the mean motions and v and PIthe coefficients of the T2 terms in the mean longitudes of the moon and thesun, respectively. The quantity K iS the ratio between the orbital angularmomentum of the earth-moon system and the rotational angular momentumof the earth. The present value of K is 4.82. The minimum value of theratio v/vP is 5.0, if N1 is negligibly small compared with N. On two differentassumptions Jeffreys obtains N/N1 = 5.1 and 3.4, respectively. Thesegive 6.3 and 7.2 for the corresponding values for v/vl. With +5!22 and+ 1' 23 for the coefficients of T2, V/Ip = 4.24 is below the theoretical mini-mum. The provisional values found in this discussion, +2!2 and +101,with v/lv = 2.18 emphasize the contradiction.

Dr. Harold C. Urey," in his Silliman Lectures given at Yale Universityin April, 1951, proposed to modify Jeffreys' formula by introducing as anadditional unknown the secular changes in the moment of inertia aboutthe axis of rotation. The formula then may be written

K -3 N1 QdCV K N N dt nvl N,N1 Q dC n1

N N dt

If numerical values are introduced including v = +2.22 + a, v = + 1.01+0.0748 a, the result becomes

Q dC N,- -0.255 + 0.055 a.N dt N

Now a = 0 4 5.5, whence

N dC N -0.255 4 0.30.

With N/N, = 5.1,

Q dCN = -0.45 4 0.30;

with N/N1 = 3.4,

Q dC= -0.55 4 0.30.

N dt

A secular decrease of the moment of iuiertia about the axis of rotation istherefore indicated. This is in agreement with Urey's views on the originand constitution of the earth.Acknowledgments.-The study presented in this paper was undertaken

VOL. 38, 1952 11

ASTRONOMY: D- BROUWER

as a direct consequence of Professor Harold C. Urey's Silliman Lectures atYale University in April, 1951. Several questions that arose in discus-sions following these lectures led me to examine the problem of the fluctu-ations more fully than I had had occasion to do previously.

I am particularly indebted to my associate at the Yale Observatory,Dr. A. J. J. van Woerkom, who assisted me with numerous calculationsand who permitted me to make use of some of the results of the experimentswith the accumulation of random numbers which we undertook jointly inan effort to find answers to various problems that presented themselves inthis investigation.About 17 years ago, Dr. Theodore E. Sterne, now at Aberdeen Proving

Ground but then at the Harvard Observatory, was working on cumulativeerrors in periods of variable stars. During a visit to the Yale Observatoryhe brought up the question whether the fluctuations in the earth's rotationmight be explained as the accumulation of random errors. As I rememberour discussion, we agreed that the evidence strongly favored large suddenchanges, and his suggestion was not further examined.A paper giving details of the derivation of the data on the fluctuation

curve is being prepared for publication in the Astronomical Journal. Ajoint paper by Dr. van Woerkom and myself, also to appear in the Astronom-ical Journal, will be devoted to our experiments with artificially con-structed fluctuation curves.

Sir Harold Spencer Jones has remarked that the fluctuations in theearth's rotation resemble observed errors in pendulum clocks that wereshown to be affected by frequent small erratic changes. In 1949 hewrote :14 "It may prove however, that the earth itself is rather likea pendulum in its behavior and that its rate of rotation is liable tofrequent and small irregular changes, so that we can at present merelyobserve their integrated effect." This view is supported by the evidencepresented in this paper.

' Stoyko, N., C. R. Acad. Sci., 205, 79 (1937).2 Finch, H. F., Monthly Notices, R. Astr. Soc., 110, 3-14 (1950).3 Taylor, G. I., Phil. Trans. R. Soc., A 220, 1-3 (1919).4Jeffreys, H., Ibid., A 221, 239-264 (1920).6 Munk, W. H., and Miller, R. L., Tellus, 2, 93-101 (1950).6 de Sitter, W., Bull. Astr. Inst. Neth., 4, 21-38 (1927).7Spencer Jones, Sir Harold, Ann. Cape Obs., 13, part 3 (1932).8 Spencer Jones, Sir Harold, Monthly Notices, R. Astr. Soc., 99, 541-558 (1939).9 Clemence, G. M., Astr. Papers Am. Ephemeris, 11, part 1 (1943).

10 Newcomb, S., Ibid., 9, part 1 (1912)."Fotheringham, J. K., Monthly Notices, R. Astr. Soc., 87, 154 (1926).12Jeffreys, H., The Earth, Cambridge Univ. Press, 2nd ed., p. 256 (1929)." Urey, H. C., Geochimica et Cosmochimica Acta, 266 1, (1951).14 Spencer Jones, Sir Harold, Ann. Rep. Smithsonian Inst. 1948-49, p. 201 (1950).

12 PROC. N. A. S.