LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class THE INEQUALITIES IN THE MOTION OF THE MOON DUE TO THE DIRECT ACTION OF THE PLANETS CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, C. F. CLAY, MANAGER. : FETTER LANE, E.G. : 50, WELLINGTON STREET. 3Letp>i|j: F. A. BROCKHAUS. Jftefo Sorfe: G. P. PUTNAM'S SONS. Bombag anti Calcutta: MACMILLAN AND CO., LTD. [All Rights reserved.} THE INEQUALITIES IN THE MOTION OF THE MOON DUE TO THE DIRECT ACTION OF THE PLANETS AN ESSAY WHICH OBTAINED THE ADAMS PRIZE IN THE UNIVERSITY OF CAMBRIDGE FOR THE YEAR 1907 BY ERNEST W. BROWN Sc.D. F.R.S. (i PROFESSOR OF MATHEMATICS IN YALE UNIVERSITY CONNECTICUT SOMETIME FELLOW OF CHRIST'S COLLEGE CAMBRIDGE AND PROFESSOR OF MATHEMATICS IN HAVERFORD COLLEGE PENNSYLVANIA CAMBRIDGE at the University Press 1908 Astron. Cambrilrgt : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. TO GEORGE HOWARD DARWIN AT WHOSE SUGGESTION THE STUDY OF THE MOON'S MOTIONS WAS UNDEETAKEN BY THE AUTHOR AND WHOSE ADVICE AND SYMPATHY HAVE BEEN FREELY GIVEN DURING THE PAST TWENTY YEARS THIS ESSAY IS GRATEFULLY DEDICATED a 3 173963 CONTENTS. PAGE INTRODUCTION . ix GENERAL SCHEME OF NOTATION SECTION I. THE EQUATIONS OF VARIATIONS 3 Numerical form of the equations 6 The variation of the moon's true longitude . 11 Abbreviation of the formulae 12 Final form of the abbreviated equations 14 SECTION II. TRANSFORMATION OF THE DISTURBING FUNCTION .... 16 SECTION III. DEVELOPMENT OF THE DISTURBING FUNCTION 26 Leverrier's expansion of I/A . . . . . . . . . . 26 Calculation of the functions $ p . 28 Solution of a difficulty 34 Calculation of the coefficients Mi and of their derivatives . . . . 35 SECTION IV. A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS . . 37 Construction of the sieve 41 Orders with respect to e, k, associated with the lunar arguments . . 45 Method for finding the periods to be examined ...... 45 SECTION V. AUXILIARY NUMERICAL TABLES 49 Adopted values for the constants of the solar system 49 Tables for A p , p , ..., for each planet 50 The values of M t and of their derivatives 61 An example of the computation of a primary coefficient .... 65 The terms retained by the sieve for computation ...... 67 SECTION VI. THE INEQUALITIES IN THE COORDINATES 70 Terms in the moon's longitude 70 Terms in the moon's latitude 87 Terms in the moon's parallax 88 Terms in the mean motions of the perigee and of the node ... 88 ADDENDUM. FINAL VALUES FOR THE TERMS IN LONGITUDE .... 89 ERRATA 93 <,!-* -^-'-tt YJ55 OF THE " { UNIVERSITY ) OF INTRODUCTION. THIS Essay aims at a complete calculation of the effects produced by the action of a planet on the motion of the moon under the following limitations and conditions: (1) The problem of the motion of the moon under the action of the sun (supposed to move round the centre of mass of the earth and moon in a fixed elliptic orbit) and the earth, is considered to have been completely solved. (2) All the bodies are supposed to attract in the same manner as particles of masses equal to their actual masses and situated at the centres of mass. (3) All the planets are supposed to move in fixed elliptic orbits, i.e., the effect of the action of a planet transmitted either through the earth or through another planet is neglected. ' (4) Perturbations of the first order with respect to the ratio of the mass of a planet to that of the sun are alone calculated. (5) The exception to the above limitations occurs in the periods of revolution of the apse and node of the moon's orbit. These periods are not exactly those arising from (1) but they are the observed periods or, what amounts to the same thing owing to the close agreement between the observed and calculated periods, the periods after all known causes have been included. The point is only of importance in terms of very long period. (6) All coefficients greater than 0"'01 in longitude, latitude and parallax have been obtained. Many are also given which are less than 0"'01 whenever they have been accurately calculated. There are, in addition, classes of terms of short period which run in series and which in the aggregate will add up at certain times to much more than 0"'01 : these have been found to be 0"'002. X INTRODUCTION (7) The maximum period considered is 3500 years, but as the sieve in Section iv. retained a few terms of longer period, these were also included in the general scheme. The methods here adopted have been constructed mainly to overcome the difficulties which have in the past prevented an accurate computation of the long period terms. They were, however, found to be equally useful for finding the terms which have periods of a year or less. These difficulties include the development of the parts of the disturbing function which depend on the coordinates of the earth and planet; the accurate calculation of the derivatives of the moon's coordinates with respect to n, the moon's mean motion ; the uncertainty arising from the possible omission of terms of long period ; and the frequent appearance of small coefficients as the difference of two large numbers. In Section I., the equations of variations for the lunar elements have been recomputed with the use of a semi- canonical system of elements. The equations for the ordinary system of elements were first given by G. W. Hill and independently, though at a later date, by S. Newcomb ; they were recalculated in Hill's form by R. Radau. In the present system, errors arising from the slow convergence of Delaunay's series have been avoided ; in fact his literal expressions have only been used in small terms where derivatives with respect to n were required but where the maximum possible error could make no difference in the final results. In Section II. Hill's method of dividing the disturbing function into a sum of products in which the first factor of each product is independent of the lunar coordinates and the other of the planet's coordinates, is exhibited as part of a general theorem. By referring these coordinates to the true place of the sun's radius vector, I have obtained the first factors directly from the expansion of the inverse first power of the distance between the planet and the earth (I/A) and of its derivatives with respect to certain of the elements of the earth and planet. Only one expansion is therefore required for all these factors, namely, that of I/A, and this has been given by Leverrier in a literal form in powers of the eccentricities and mutual inclination. The expansion also contains the coefficients in the expansion of 1/A (the value of a'/ A when the eccentricities and inclination are zero) and its derivatives with respect to the planet's mean distance: the formulae for finding the coefficients and their derivatives are here put into forms which admit of rapid and simple computation. The factors containing the moon's coordinates, together with their deriva- tives with respect to all the lunar elements except n, are found from the results of my lunar theory; a special method which I gave five years ago and which does not require the use of literal series in powers of ri/n has been INTRODUCTION XI used to find the derivatives with respect to n. These methods for finding the planetary and lunar factors are set forth in Section in. The only one of the difficulties previously mentioned which has not been considered up to this point is the danger of omitting long-period terms with sensible coefficients. In Section IV. formulae are constructed" which permit one to find rapidly an upper limit to the magnitude of any coefficient. By means of them, all terms having coefficients greater than 0"'01 and periods less than 3500 years have been sifted out; there are about 100 of such terms, excluding the terms of short period for which no sieve was required. Sections V., VI. consist of numerical results. It may be noted that the values of A p , B p , ... are also required for finding the perturbations of the earth by the planets and of the planets by the earth, while those of Mi and of their derivatives (as well as the equations of variations contained in Section I.) are available for the computation of lunar perturbations other than those due to the direct action of the planets. No new inequalities sufficiently great to account for the observed dis- crepancy between theory and observation appear from the direct action of the planets, as shown in the tables of Section VI. Radau's well-known list of terms in longitude has required considerable extension as far as the short period terms are concerned, and a few new long period inequalities with small coefficients have been computed. The more extensive developments of this essay have shown that some of his coefficients require alteration, but there is a general agreement for all those portions which he has taken into account. Only a few slight verbal changes and corrections of errors in copying have been made to the first five sections, with two exceptions mentioned below, since the award of the examiners. But I have gone over all the computations for finding the short period terms and the larger long period terms during the year that has elapsed and have made the following corrections to the results in Section vi. : Argument Former coefficient Corrected coefficient 1+3T-WV+33 - 0"-35 + 0"-35 -1-WT + 187-151 -15"-22 -14"-55 Z+29T-26F+ 112 + 0"-117 + 0"-108 2D-J + 21T-20F-87 + 0"-111 + 0"-126 2l-2T> + GM-5T+2ir-l + 0"-040 - 0"-038 together with their accompanying short period secondaries. The signs of those coefficients containing h on p. 86 have been changed. The annual mean motion of the perigee has been altered from 2"'66* to 2"-69. * I gave this value in a paper referred to on page 3 below. xii INTRODUCTION The values of M 2 and of its derivatives with respect to n, e, k under the argument on page 61 have required a factor 2. This error necessitated slight changes in some of the coefficients whose primaries were independent of the lunar angles ; the largest correction was one of 0"'019 in the coeffi- cient of sin (T - F). A wrong sign in the computation of the equations of variations (see Errata at the end of the volume) gave rise to a few almost insensible changes in certain coefficients. The additions are : Argument Coefficient Of-27 T +63 -0"-012 2l-2V + 4(M-T)-l +0"-017 21- 20 + 8^-67*+ 63 -0"-019 -l -0"'031 No other change or additional coefficient has been greater than 0"'010. The Addendum containing the results obtained by adding together terms of the same argument in Section VI. is also new. During the summer of last year Professor Newcomb's new work* on this subject was published. His methods differ so completely from those given here that no comparison is made easily except in a few of the final results where the indirect action is insensible or is separated from the direct action. For the large inequality due to Venus he obtains a coefficient of 14"'83 while mine is 14"'55 ; a portion of the difference is probably due to certain terms of the second order relative to the ratios of the masses of Venus and of the earth to the sun which Professor Newcomb has included. On the other hand, he states that the possible errors arising in his method may be of the order of this difference, while such errors are excluded from my result. His results and mine for the annual mean motions of the perigee and node agree within 0"'01, which is the limit of accuracy to which I have obtained these quantities. * "Investigation of Inequalities in the Motion of the Moon produced by the Action of the Planets." Carnegie Institution, Publication 72, Washington, D.C., June, 1907. E. W. B. NEW HAVEN, CONN., U.S.A. 1908 March 27. OF THE UNIVERSITY OF GENERAL NOTATION. THE axes of x, y are taken in the ecliptic of 1850'0 and the centre of mass of the earth and the moon is supposed to describe a fixed ellipse around the sun in this plane. As it is more convenient to use the motion of the centre of the earth than of this centre of mass, a slight well-known change, noted below, is necessary in the disturbing function. The axis of x is parallel to the line joining the earth and the sun. In the scheme of notation which follows, two sets of constants are given for the mean distance, eccentricity and sine of half the inclination of the moon's orbit. The first set is that which I have used in the expressions for the rectangular coordinates of the moon ; the second set is that of Delaunay in the final form which he gives to the expressions for the longitude, latitude and parallax. The longitudes of a planet, of its perigee and of its node are as usual reckoned along the ecliptic to its node and then along the orbit. True long. Mean long. Mean anom. Mean long, of node Mean motion Mean distance Eccentricity Sine half inclin. Coors., origin earth sun Moon V 1 = l + g + h n a, a e, e k, y x, y, z, r Earth V T l' T-as' Planet V" P l" = P w" h" flP at a" e" y" , rj, , A x", y", z", r" dw 1 n = mean motion of the moon = -j-= , & 2 = its perigee = ^ , b *= node ^' BR. 2 GENERAL NOTATION d, c 2 , c 3 are the canonical constants complementary to w lt w 2 , w 3 after the problem of the moon's motion as disturbed by the sun, supposed to move in a fixed elliptic orbit, has been solved. R = the disturbing function of this problem, arising from the direct attraction of a planet. The symbols for the mean longitudes of the planets are : Mercury, Q ; Venus*, F; Mars, M\ Jupiter, /; Saturn, S. * No confusion will be caused by the use of the same symbol for the true longitude of the moon and the mean longitude of Venus. The notation of Eadau has been adopted with a few changes. SECTION I. THE EQUATIONS OF VARIATIONS. LET w^ w 2 , w 3 represent the mean longitudes of the moon, of its perigee and of its node, and suppose the problem of the moon as disturbed by the sun, has been solved. Then it is well known that if a disturbing function R be added to the force function of the moon's motion, the change in the latter due to R can be obtained by solving the equations dd dR dwi dR . where 6 lf b 2 , b 3 are the mean motions of the moon, of its perigee and of its node, and c l} c 2 , c 3 are the canonical constants corresponding to w ly w 2 , w s ] the Ci are functions of the arbitrary constants n, e, k of the moon's motion, and they also contain the constants n', e', depending on the sun's motion. The substitution of the new values of d, Wi, thus found, in the expressions for the coordinates will give the disturbed position of the moon at any time. The constant part of R only gives constant additions to the bt, i.e., to the mean motions* : this part will be neglected, since it has no effect on the new terms to be found. Hence & t = n. Change to the semi-canonical system n, c 2 , c 3 , retaining the Wi un- changed. Putting ~~7 ^ ^ Pi dn and remembering that ,.. , dci _ db 2 dCi _ db 3 db 2 db 3 dc 2 dn ' dc 3 dn ' dc 3 dc 2 ' * I have found these changes in an earlier paper : Trans. Amer. Math. Soc. , Vol. v. pp. 279 284. A fresh computation just made gives 2" - 69, -1"'42, for the mean motions of the perigee and node respectively. The former is 0"'03 more than the value given in the paper. 12 4 THE EQUATIONS OF VARIATIONS [SECT. we obtain equations (1) in the semi-canonical form dn ldR, dbz dcdb 3 dcA dw l _ 1 dR . ' dt ~ a 2 3 dn _ _ dw l dn ' dt dn ' dt ' dt ~ a 2 /3 dn dc z dR dw, dR di=^ ' w = -^ dc s dR dw 3 dR , fdw 1 , \ db in these equations b 2 , 6 3) c a are supposed to be expressed in terms of n, c 2 , c 3 and R in terms of n, c 2 , c g , w lt w 2 , w 3 . Consider any periodic term of R : R n*tfA cos (qt + q) = n' 2 a?A cos (i l w l + t' 2 w 2 + i s w 3 + q"t + q'"}, where a is the linear constant of Hill's variational orbit and of my lunar theory and A is a numerical coefficient (that is, its dimensions with respect to time, space and mass are zero) ; q"t + q" is a combination of the solar and planetary arguments. Then since qt + q' is independent of the Ci and A of the Wi, the first three of equations (2) become dn n' 2 a 2 . dq . , , ,, It will now be supposed that R contains a small factor whose square may be neglected. The coefficients in the right-hand members of the last set will H then be constants and we can integrate. Put m= , and let 8n, 8c 2 , Sc 3 iff denote the increments of n, c 2> c 3 , due to R. Then Bn ra a 2 dq n' . ,. - = - . -^ - A cos (qt + q ), n ft a? dn q (3) 8c 2 = Lm . A cos (qt + q'), r*& r* v -* * * ; 3 _ . a 2 n tf 3 ' a 2 q Again, if we put A - A ~ dA -^ cfc 2 l] THE SEMI-CANONICAL FORM the other three of equations (2) become idw, n' 2 a 2 . (3a) dt ~nft ' a*~ dw 2 _n 2 a 2 /. AI dh dt n a 2 \ ft dn dw 3 n' 2 a 2 / . A! db 3 \ ,. , -T7- = -5 (A, + -g 1 . -,! ! cos (gtf + gr') -(- 6 3 . dt n a 2 V P W Let 8b l} Sb. 2 , S6 3 , Si^, Bw. 2 , Bw 3 denote increments due to M. Then n' a 2 dq n' $b l =*Sn = -- 5 . -j 1 . 2 cos /V ^l /i /^ fif* {.4/lv \A/\S% ^*^- / 3 , a 2 n ' A ( 1 ^2 d^ 2 d6 2 2 d6 2 . , = n -- . A\ TT . i . -= H a = . i 2 + a -7 . t 3 cos = . - -- . T- . -r . 2 - a? q \ ft dn dn dc 2 dc 3 by equations (la). Putting q 2 = - no? , ~T^> ~T^ > res P ec tively. To obtain Bw 1} 8w 2 , 8w a (the increments of Wi, w 2 , w s ), it is necessary to replace Wi by w t + 8wi, and 6 t - by bi+Sbi in (3 a), to substitute for Bbi the values just obtained, and then to integrate. These operations give (4) 1 a 2 / n' A dq n' 2 A -- A ^ l -- ft a 2 V q dn q 2 a 2 f w / . A ! d6 3 \ ?i 2 . /o s 1 d6 3 dm'. Next, let s'= no. of seconds in the daily mean motion of the sun = 3548 //< 19, . s= M argument qt + q'. Then Also, put for brevity q s (5) =Z - 206265, 4 m a 2 /3 4 ra a 2 The coefficients of the right-hand members being thus expressed in seconds of arc, equations (3), (4) become (6) Sn ,,,dqA , - = -/ -^ cos (at + q), n dn s . dn > - A ft + i . sin (q t + q') ; 2 . n dn dn s \ an '] s* where, to recall certain definitions, _ 1 dci p = 5 . -j , C a 2 dn i&9. g 2 = - na 2 -i q Numerical form of the equations of variations. It remains to be seen how these quantities may be put into numerical a 2 form. In /, /' the factor is immediately obtained from Hill's results* for a m" the variational orbit ; m is a well known quantity ; T is known as soon as * Amer. Jour. Math., Vol. i. p. 249. l] NUMERICAL FORM OF THE EQUATIONS OF VARIATIONS 7 the particular planet is chosen ; thus /, /' remain the same for a given planet and ///' for all planets. The coefficients A, A lt A z , A 3 will be found later on, while s is known as soon as the particular term of R has been chosen. There remain for calculation .,_. dc-L db* dbs dbs db^_db 3 dba dn ' dn' dn' dc 2 ' dc 3 dc z ' dc a ' which, depending only on the orbit of the moon as attracted by the sun and earth, are the same for every perturbation of the moon's motion, and there- fore apply not only to the present investigation but also to all investigations where a disturbing function R is added to the moon's force function. Some idea of the degree of accuracy required is desirable. The largest known inequality is that with the argument 1 + I6T 18F, which has a coefficient of about 15". For this ^ = 1, i 2 = - 1, i 3 = 0. The principal part is given by _/ 4 .._/ 4 (!_*!}__/ 4(1+ -01486), J s 2 dn J s 2 V dn) J s 2 v There is no other coefficient which is so great as 2". Since the degree of accuracy aimed at is 0"'01, it will be sufficient to use four place logarithms and four significant figures for the functions (7) so that the final results will be accurate to at least three significant figures. But certain of the functions are only needed to one or two significant figures, as will appear immediately. The functions c ly C 2 , C 3 are the same as Delaunay's L, G L, H G after the final transformations and the changes to his final system of arbitraries, n, e, 7 have been made. As my results will be used for the calculation of the moon functions, it will be more convenient to transfer A^, A 3 to my constants e, k. Let -f* denote the derivative of a function Q with respect to n when it is CtTl expressed in terms of n, c 2 , c 3 , and jr when it is expressed in terms of n, e 2 , 7 2 . Then the following equations serve for the transformation of the derivatives of Q from one set to the other*: dQ = /dQ\ _ dQ /dc 2 \ _ dQ dn \dn ) dc 2 ' \dn ) dc s \dn - de z dc s ~de 2 dQ rdQ dQ df ' The functions considered here involve e, y only in the even powers. 8 THE EQUATIONS OF VARIATIONS [SECT. the same equations will serve for the transformation from the set n, c 2 , c 3 to the set n, e 2 , k 2 if we replace e 2 , ( s ) which is about -^ less than my value. But I have been unable to find the coefficient 3'0o76t' in 8w 1} which Radau gives : from his data I make this coefficient 3'0791 J which, reduced to my form, becomes '9964. The * American Eph. Papers, Vol. in. p. 390. t Loc. cit. , pp. 35, 36. J This apparent error seems to be due to some confusion in the substitution of the numerical values for n before and after the final Delaunay transformation. l] VARIATION OF THE MOON'S LONGITUDE 11 difference, '0036, is about the error to be expected owing to the slow con- vergence of the series from which Radau obtains his coefficient. It may be added that the value of /3, on which my coefficient of ^ alone depends in this connection, can be determined accurately to at least seven significant figures, and the value for it quoted above has been again verified Tind tested. The Variation of the Moons True Longitude. The coefficients in longitude are obtained by substituting the values thus found in (10) IF-H dV dV dV dV z dV , + -y . 8w 2 + -j . Sw s + -j- . Sn + -j . Sc 2 + -T- . 6c 3 aw 2 aw s an dc 2 dc 3 the first term of which contains the primary inequalities, and the remaining terms the accompanying secondary inequalities. The largest coefficient in V that of sin I produces a maximum dV coefficient in 8V through -j . 8c 2 , less than 1"; the coefficient of sin I in ac% V is 22640". It has therefore been necessary, in order to include all coefficients greater than 0""005, to find those terms in V having the factor e p which have coefficients greater than 100", or, if p = 0, to 400". From my results I obtain V=w 1 + [9-0405] sin I + [8-3470] sin (2D - 1) + [6-9688] sin (2D + 1) + [6'2703] sin (4D - 1) + [8-0603] sin 2D + [7-5715J sin 21 + [7-0112] sin (2D - 21) + [61738] sin (4D - 20 - [7-3001] sin 2F + [6'4273] sin (2D - 2F) - [6-7818] sin D - [6-7262] sin (I + 1') + [7'0000] sin (2D - 1 - 1') + [6'8555] sin (I - 1') - [6-3398] sin (I + 2F) + [6'2826] sin (I - 2F) + [6-2434] sin 3Z + [5 -8060] sin (2D - 3J) - [7-5110] sin I' + [6-9040] sin (2D - 1'), the coefficients being expressed in radians and the notation for the arguments being that of Delaunay. From these results the derivatives with respect to Wi are immediately obtainable with D = w l earth's mean longitude, l = W 1 W 2 , F = Wi w s . OF THE UNIVERSITY } OF 12 THE EQUATIONS OF VARIATIONS [SECT. dV For -j the results of Delaunay might be used, but it will be shown dn dV immediately that -y- Sn contributes nothing sensible to SV a result probably of the use of the system n, c 2> c 3 instead of n, e, 7. The derivative has however been found for the largest terms in order to show this fact. The derivatives with respect to c 2 , c 3 are obtained from my results with the help of equations (8). They give dV H ~cfa = + 1 8 ' 0465 ! sin l ~ t 8 ' 4669 ] sin ( 2I) - - [7-2833] sin (2D + 1) - [8-3965] sin 2D + [6'8784] sin 21 - [71761] sin (2D - 21) -[6-8271] sin 2F, dV na?^=- [11-5828] sin I - [10-8895] sin (2D -I)- [9'5959] sin (2D + 1) - [10-0018] sin 2D - [10'4147] sin 21 - [9'8546] sin (2D - 21), m <^= [9-6959] sin 2F. The latitude and parallax are treated in a similar manner. Abbreviation of the Formulae for SV. In the actual applications to the calculation of the direct inequalities much abbreviation of these formulae is possible. For example, in the large class which has its arguments independent of the lunar angles, ^ = i 2 = i s = 0, and therefore Sn = &c 2 = Bc 3 = 0. The maximum values of the remaining coefficients are fdA_l,,, s de 5 /'<^i i,. s dk 10 : 1 dV 1 <2F 1 9' m 3'5 J " dw 3 ' 250 ' . dV in -r; 1, TT awt Hence, to the accuracy desired, we can put $ Wl = /' 4l s i n (^ + q ') } Bw^ = [1 1 '0999] ^ J - sin (qt + q), 8w 3 = [10-2620] -- sin (qt + q'). CvK S When i l} i zy i 3 are not all zero, it is still possible to limit the formulae very materially after Bw l has been found, owing to the limitations on the magnitude of the coefficients in Sw t and V. For this purpose we note that, for the greatest coefficients, dV dn dV na 2 dw l -1 9' dV dc 2 < 40, no? dF C?C S dV <1 <2F dw-i dw 3 250' ABBREVIATION OF THE FORMULAE FOR 8F 13 Put \ = - 4 + -014864 - -0037444, c= x /4, f^ = [5-32390]. Then (11) = [5-3239] Cs cos (qt + q')> -1 = [4-8419] ^ 4 cos (qt + q'\ no ^ = [4-8419] 4 cos (qt + q'}. no? A-i Also, if \2 = -014864 - -0070664 - '0081484, A, = - -0037444 - -0081484 + -0012104, dA _. . , dA = . A the equations for Swi become = - [3-4959] f . 4 1 + [7-3842] J 2 - [4-6150] f 7, + M (7 sin (^ + q'\ AI AJ . {+ [2-8972] f . 4- 1 + [5-9349] fj 3 Aj -. Aj - [4-5805] f J 2 + M sin (qt + q'). Aj ' AX) Now Hence < 5000", (7s < 10000". Aj Therefore if the limits of the coefficients to be considered be 0"'003, dV Sn and the parts contributed by the first and third terms of Sw 2 and dn &w 3 are insensible ; as a matter of fact no one of these parts is so great as 0"'001. Hence we obtain the final form of the equations of variations given on the next page. It is necessary to remember that, although the derivatives with respect to c 2 , c s have been transformed into derivatives with respect to e, k, the derivative with respect to n is to be taken on the assumption that the coefficients are expressed in terms of n, C 2 , c 3 . THE EQUATIONS OF VARIATIONS [SECT. Final form of the abbreviated Equations of Variations. (12) sn 8w = {[7-3840] f J 2 + M C^i * lj where = [6-9349] J 3 + (7 sin (3* + 4> 4 are the coefficients of w 1} w 2 , w 3 in qt + q', A n ^ ( A i\ aA ' A \r ^A One or two particular cases of frequent occurrence may be mentioned. In all cases C ^^ . For multiples of the arguments given, divide the second term in 8w l by the multiple. The argument is the moon portion of qt + q. The omitted terms are either zero or negligible. Arg. 0. Here 4 = 4 = 4 = 0. f'A, Sw, = J - - sin (qt + q), o 7/1 Pi 1 *OQQQ1 - Kin \JWn I J. X \JU*J t/ I A ^ C/M/1 . J A de r - [10-2621] Arg. l = w,- &i; 1= jl -[5-3175] *- 8w 2 = {- [7-3776] s - "02161} C sin (qt + q'\ ^ = [4-8355] Cs cos (qt + q'), 8w 3 = - [6-9285] ; 3 Cs sin (^ + q'} (for the latitude only). l] FINAL FORM OF THE ABBREVIATED EQUATIONS OF VARIATIONS 15 Arg. 2D - I = w, + w z - 2T. 8 Wl = \l - [5-3304] s 4-4 C sin (qt + g'), ( A ) Sw, = - [7-3905] sC sin (qt + g'), -^ ! = _ [4-8484] sC cos (qt + q'\ no? Arg. 21 2D = 2w 2 + 2T. 8w, = {l- [6-8505] a- ( 8w 2 = {- [9-2119] s - -4753} (7 sin ^ = [6-6698] C7 cos (qt + g'). na 2 There is quite an extensive class of terms containing this argument in which 8w 1 is insensible. For these we can put 8w 2 = [5-7642] -f- . 4 sin (qt + q'\ s e = - [3-2221]^ . 4 cos (at + g')- no? J s e 2 The great majority of terms in R to be considered contain powers of the lunar eccentricity as a factor and the principal terms in the moon's true longitude have the same property. In nearly all such cases it is permissible to neglect higher powers of the lunar eccentricity and inclination. When dV these two conditions are satisfied it is not necessary to find Sc 2 , -j . For then j z = | i 2 and the ratio of the first term of the coefficient in 8w z to dV dV that of 8c 2 is the same as the ratio of the coefficients in e -3 , j . This f arises from the fact that Bw 2 depends mainly on e -j- , while 8c 2 depends on cte -= . If then dw z Sw 2 = Q sin (i 2 w 2 + i/r), V = Q' sin (i 2 w 2 + ^') , where i|r, ^r' are independent of w 2 , and i 2 , i 2 have the same sign, dV dV -r Bw 2 + -j Bc. 2 = QQ'i 2 sin {(i, - i 2 ) w^+^ ^'}. It is to be noted that Q is here the first term in the coefficient of Sw 2 . An exactly similar theorem holds with reference to c s , w 3 and the terms which contain powers of k as a factor. SECTION II. THE TRANSFORMATION OF THE DISTURBING FUNCTION. IT is well known that the disturbing function for the direct effect is R, where m [(l-^ + ^- In order to take into account the motion of the earth round its centre of Q mass, the terms containing are as usual to be multiplied by the ratio of ct> the difference and sum of the masses of the earth and moon. a a a a *87 + ^ + *aT = aQ ; then, by Taylor's theorem, as) _ i __ = [1-! i/4,Y-iY-iY+ I 1 ' + - " 1 ri /ay ~u l\2\dQJ ' JA' Since R will enter only through its derivatives with respect to the lunar elements which are not present in A, we obtain The separation of the terms of R into sums of products, one factor involving the lunar coordinates and the other the planet's coordinates, suggested by Hill* and adopted by Radauf, is implicitly used here, and the division is the same in principle, but it will take a different form with the use of complex coordinates, and the method of separation is made on a general plan which can be applied to terms of any order in R. * Amer. Eph. Papers, Vol. in. t Ann. de I'Obs. de Paris (Mm.), Vol. xxi., 1892. SECT. Il] SEPARATION INTO SUMS OF PRODUCTS IT Put as + y V- 1 = u, + 77 V- 1 = u^ , x y V 1 = s, 77 V 1 = 8!; then A 2 = Uj s x + 2 , r 2 = us + 2 2 , a _ _9 _9_ _a_ -~i/~k ^ o "T^-v ~r ^ O4 9Q 9uj 9sj 9 And since I/A is a solution of Laplace's equation with respect to , rj, 9 2 1 / 9 2 9 2 \ 1 9 2 1 _ [ I I n, . "i Cxo A ~ I ~\ i-o ~ ^ a I A " " ' 9 2 A V9f 9W A da l ds 1 A ' Hence, omitting I/A for the sake of brevity, 1 / 9 \ 2 1 f 9 \ 2 1 / 9 \ 2 9 2 1-us 9u 1 9s 1 l 9 /9\ 2 , 9/9 9\ o ^ I oP ) +*^ U 5 + s ^~ 2 w/ 9?v ou, a = real part of (r 2 - 3* 2 ) 55- + u 2 L- + 2^u / 9u 1 9s 1 T = real P art of - ( r * ~ 5 * 2 ) u aT - 3 V9Q/ y 9u 1 2 9s 1 3 / 9 \ n and so on. The method is the same throughout : expand (53) and replace 9 2 \ m 4 5 ^ and us by r 2 -^ 2 . I now give a method for replacing the derivatives with respect to the coordinates, by derivatives with respect to a', T (the earth's mean longitude), h", y". In this way it is possible to make use of Leverrier's expansion* of I/A in terms of the elliptic elements of two planets to the seventh order inclusive with respect to their eccentricities and their mutual inclination. Expressing the coordinates in polar coordinates, we have _/+(!_ 7 " 2 ) r " cos ( V" - V'} + y" 2 r" cos ( V" + V - 2A"), (15) 1?- (1 - 7 //2 ) r" sin ( V" - V'} - y" 2 r" sin ( V" +V- 2A"), = 2 7 " Vr^/V' sin ( V" - h"). Whence A 2 = r" 2 - r' 2 - 2r', Ul = - r 1 + (1 - 7 " 2 ) exp. ( V" - V) V^Tl + yV exp. (- V" - V + 2h") the expression for Sj being obtained by changing the sign of V 1. * See below. BR. 18 THE TRANSFORMATION OF THE DISTURBING FUNCTION [SECT. From these results we obtain a a dr' 3u, 3s, ' (16) 4 -* a U , U I r = r , + v 1 3v dr ov ouj osj os 1 d , d / ^ 3 3^3 , 3 3w dr' dV' 1 du 1 1 3sj 3u x ' PJ o the notation ^- , r being introduced for a few pages for brevity. Let F denote a function of f and of the product UjSj only. Then / 1 fx Q \ __ tyty* 9/y*' v i0dl / A ^ ' a,,,~ ^ r ^7 > (17) Hence 8v3w i r/vAVa.Y 8 vi 1 A [I 3^ laF'^jA' Af_2r' - 3v I 3u x ' A = 4/2 A av jas! * A ' a 1 A av A Similarly, A A ' and the general law is evident. The derivatives with respect to u l5 Sj are thus expressed in terms of derivatives with respect to r, V , when for ^ o * . x have been substituted their values (16). dv 3w Next, A 2 = r' 2 + r" 2 - 2r'r" (1 - y /2 ) cos ( V" - V) - 2r'r"y" 2 cos ( V" + V - 2A") = / 2 + r " 2 _ 2r y cos ( V " - V ) + 4rV V /a sin ( F" - A") sin ( F' - h"}. f_d_ J_ 3 ^ 1 _ 2 (1 - 7 //2 ) r'r" sin ( F 7/ - h") cos ( F' - h") J^ 2(1- 7 //2 ) rV /7 sin ( V" - h"} sin ( V - h") 7 ) 8 7 "" 2 A~ A 3 Il] EXPRESSION BY DERIVATIVES OF I/ A 19 Therefore - " 9 7 " 2 A 2(1- 7 //2 ) r'r" sin ( V" - h"} exp. - V^I (F^ - A") This gives 9 1_ _ 7" exp. \/^ ' A A (As ^/A 3 is real, we incidentally have the relation [sin (V- h") . (A, + 2 -i /a ^) - cos ( F' - *) (1 - 7 "-) ^] 1 = 0, and the sign of the V 1 may be changed.) Since /A 3 involves u u Sj in their product only, the formulae (16 a) are available ; and as d exp.V~l(7'-r) exp.W^l(F'-/Q/a o - 7 - * - 7 - 1 o -- * 9v r r \9v 9 exp. \/^l ( V - h") -, exp. + V^I ( F' - h") d v ^ > Jb = -, - ^ n , dw r r ow we obtain, by substitution of the various results, (18) B . C. 2 exp - + >J~l (1- 1 3 9 /8 \/3 3 U 9wV9w~ A9w~ V A 22 20 THE TRANSFORMATION OF THE DISTURBING FUNCTION [SECT. The derivatives with respect to v, w are easily expressed in terms of deriva- tives with respect to the elements by means of the well known formulae ,d_ , d_ J_ d_ r dr'~ a da" dV'~dT' where T is the mean longitude of the earth and where it is supposed that the angles referring to the earth are expressed in terms of T, I'. Also, Leverrier's development depends on a = , where a^ is the mean distance of the inner Gvg planet, a 2 that of the outer. Hence if we put a a d~a ==D we have, since -r- is a homogeneous function of degree 1 with respect to a, a", a x = a", 0-2 = a', r =-, -, = a - -- 1 = D 1 for inner planets, or da ?) 7\ (h = a, a 2 = a", r' =-, = a - = D for outer planets. or oa. The planet portions of the disturbing function are therefore expressed in terms of derivatives of I/A with respect to a, T, h" , y" 2 , the quantities present explicitly in Leverriers development. Each term in the development of R consists of two main parts together with a constant factor. The first part is a function of the moon's coordinates alone (with which will be combined the factors 2 , 3 , e ( v '~ h ")^ - 1 J ; the second part consists of the derivatives of I/ A which is a function of the earth's and of the planet's elements only. Let Q denote an angle present in the first part, an angle present in the second part, that is, in I/ A, so that 6 is the argument of a term in R which is under consideration. Put j for V 1, and* COS . - = i (M, + M 3 ) e + *(#,- #) e ~ j = M * cos e +J M * sin I V Qi n'* ri npiCy-k") \- -= M^ 6 f = i/ 4 cos + JM 4 sin 6. 1* t These functions are calculated once for all and serve for all the planets. * The exponent always attached here to the exponential e prevents any confusion with the same symbol used to represent the lunar eccentricity as defined by Delaunay. t Owing to the absence of h" from z, u, r', the part e~ j6 is not present here. II] FORMULAE FOR THE PRINCIPAL TERMS 21 For the second part put* ^ = P cos = P cos (iT+ 2i,h" + '), where is independent of T, h". Also for inner planets P 4 cos + JP 5 sin = - D + 1 + j - so that The first line of R, which consists of those terms in (18) which have the m factor -T 75 , becomes with these substitutions, after inserting the unit factor*!* ** (19) E 1 = fn /2 a 2 47J2- - J= M, (P 4 P.) cos (0 7 either all the upper signs or all the lower signs being taken according as it is convenient to use the sum or difference of the angles 6, . It will be noticed that the P p contain the divisor a which will cancel the a outside the parentheses ; otherwise they contain a', a!' only in the form a' fa" = a. * The letter i has also been used elsewhere in a different connection ; but no confusion need occur. t The denominator is properly the sum of the masses of the earth, moon and sun ; but the difference is insensible. 22 THE TRANSFORMATION OF THE DISTURBING FUNCTION [SECT. For exterior planets the same formulae hold if (1) we substitute D I for D, (2) replace the factor a by aa", since Leverrier's formulae are then to be expressed in the form -77 func. (a). CL o For the second part of R. which involves the factor and which will be a denoted by R z , put for the moon portions, n '3 n (& _ P^2\ ^ ; = \ (M 6 + M 7 ) e* + i (-flf. - M 7 ) tr* = M 6 cos +JM 7 sin 0, J *- - ( r * ~ K> = M " e6j = M ( cos e +J sin ^X = M lz (cos +j sin 6), and for the planet portions, in the case of the inner planets, = \3w 3 +j [(D+ I) 2 -t 2 P 6 cos - jP 7 sin = I = -- 2 ] 5-5- . P cos r J r P 8 cos - ?P 9 sin = ( -- 4] ( = -- 2 ) ^ P cos 17 r r P 10 cos + JP U sin = j - 2 P 12 cos + JP 13 sin = j - 9w WKM Hfc UNIVERSITY i 'FOR* II] FORMULAE FOR THE PARALLACTIC TERMS 23 so that P S P 7 = - {(D+ I) 2 -; 2 } (D + 3 i}P, P 9 = - (3D 2 + 18 D + 23 + i 2 } iP, Pi. PU = - {(D + 3) (D + 1) - * + 2i} (1 - 7 " 2 P. Then (20) .R, = -L ^ . W ' 2 a 2 * a' Filf 6 P 6 + jf r P r + M 8 P 8 + M 9 P 9 J. f tYL CL os (0 , . , 10 VI y The changes for outer planets are the same as in the case of jR x . The moon coordinate u is-referre.d to the true place of the sun. As my results will be used to calculate the functions of u, s, z, and as these results are referred to the mean place of the sun, it is necessary to replace u by u e~^ ' where u is the same as the u of rny Lunar Theory. The formulae for M p and P p are collected and slightly altered in form at the end of this Section. In appearance they are somewhat complicated, but they are easy and rapid for numerical calculation. For example, the effect of the first factor in P 4 , P B , P 1() , P u , P 12 , P 13 on the expansion of I/A is seen at a glance; it is shown below that where P x has not been tabulated it is rapidly found from a general formula; and the effect of D on P or PJ requires nothing more than the replacing of K p by (p + l)K p+1 +pK p . The great majority of terms involve only P 1} P 2 , P 3 , while the functions P 6 , P 7 , ... are required only for the very few sensible terms which have o the factor . a In the above development it is assumed that the elliptic values of , 77, are used. If the effect of their perturbations is to be included, the new part SR of H can be found by operating on its value (14) with where 8%, 8rj, 8 are the disturbed values (supposed known) of , 17, Similarly, x, y, z represent the coordinates of the moon as disturbed by a sun moving in a fixed elliptic orbit. If Sx, Sy, Sz are the perturbations produced in x, y, z by the value of R in (18). we can include the effect of 24 THE TRANSFORMATION OF THE DISTURBING FUNCTION [SECT. these additions to x, y, z (perturbations of the second order relatively to the masses) by a further disturbing function S'R=Sx 8 Bz - dx */ dy dz ' but then the w t , Ci must be newly defined, that is, they require slight additions probably insensible in the equations of variations. There is no difficulty in the calculation of these two new functions. They are obtained easily by the method given above. It will be found that SR, for the first term of R, involves the functions P 6 , ..., P 13 instead of P!, ..., P 5 , while the expansion of S'R, which involves changes in the M p only, is so simple that its value is obtained immediately. The formulae to be used are collected below. The Functions M 1} M 2 ,.... a '2 J& _ 3^2 M l = coef. of ei Q in % . - f 2 o2 / d i ( M 2 4- M 3 ) = coef. of e^ 6 in -^ r- a z if 4 = coef. of e je in -?- tf &-*>") ,-S3- Jlf 7 ) = coef. of e^ in e** ^''^ . " r A of = coef. of = coef. of ei e in f- r 3 = coef. of &* in r r 3 Functions P l} P 2 , ____ - = 2 P cos < = SP cos (iT + ^h" -A Inner Planets. 2 P 13 = - [(1 - y" 2 ) g4 (i + ^)] [A + (2 + 2i) DP + 2(i+i? P]. Il] COLLECTION OF FORMULAE TO BE USED 25 Outer Planets. n P* = ~ [(1 - y" 2 ) g4s ( + ^s)] [A + ( 2 - 2) The Disturbing Function is E t + R 2 where i^ has the value given by (19) on p. 21 and R 2 the value given by (20) on p. 23. Owing to the fact that M 2 is nearly equal to + M 3 and P 2 to + P 3 in most cases, the portion M 2 P 2 + M Z P 3 of B^ was usually put into the form for computation ; the degree of accuracy of the tables in Section V. was then always sufficient. SECTION III. DEVELOPMENT OF THE DISTUEBING FUNCTION. Leverrier's Expansion of -r-. LEVERRIER* expands in powers of the eccentricities of the two planets and the square of the mutual inclination of their orbits. The arguments are their mean motions, the longitudes of their perigees measured along the fixed plane (ecliptic at a given date) to the nodes, then along the orbits, and the longi- tudes of the nodes. As one of the planets in the present investigation is taken to be the earth, and the fixed plane the ecliptic, supposed immovable, Leverrier's arguments become in the present notation, for an inner planet, T = T ' = h", l'=T, a> = T-V, X = P, &' = P - I". (Leverrier denotes the mean longitude of the earth by I' ; I use this letter for the mean anomaly.) For the coefficients I use the same notation with the exception of 17, which I denote by 7". For an outer planet the accents (except in the case of h", 7") interchange. Put A 2 = 1 + a 2 - 2a cos (T - P), then with Leverrier, using S to denote summation for integral values of i from + 00 to oo , I put i(T- P), a A? a 2 = I a' 2D * cos i (T - P), * Annales de I'Obs. de Paris, Vol. i. The terms of the eighth order have been computed by Boquet, ib. Vol. xix. SECT, in] LEVERRIER'S EXPANSION OF I/ A 27 and* ~ =ia'23 or, in general, If K (i) be any one of these coefficients, put Then, when the functions {3 s , P (i) have been calculated, we have all the materials for obtaining the functions P p . Leverrier, besides the notation just given, uses certain functions of A, B,..., in the development. The notation adopted here is the same as his. It is (21) = 15 Of these, it has been rarely necessary to use any but E (i) , L (i} , G (i) . As powers of the A (i) , B (i} do not occur in any of the formulae which will be used, the brackets round the i in the index will be omitted. For brevity we shall put and when the index i can be omitted in any equation without causing confusion, it will be dropped. The coefficients in Leverrier's expansion of I/A are all functions of e', e", 7" and the A p , B p , .... The method outlined here for deriving all the planet functions from this one expansion does not necessarily give the shortest algebraical expressions for the coefficients, but for numerical computation, which is the principal end in view, these expressions have this advantage that they require very little use of logarithm tables. The calculations consist mainly of additions, subtractions and multiplications by integers less than 100, and the functions are read straight from Leverrier's expansion. Moreover, * The reversed letters 3, ~"\ are used to distinguish from other functions defined on this page ; Leverrier does not need the two last expansions. 28 DEVELOPMENT OF THE DISTURBING FUNCTION [SECT. it is possible to see almost immediately when the terms in a given coefficient become insensible. For example, in the case of the great inequality due to Venus, argument 1 + 16T 18 F, the principal term has the factor 7 //2 and argument 1+ 16T 18F 2^", and the terms of order y" 8 are just sensible. The following table of the coefficients actually calculated shows immediately where the calculations can be stopped. The first column gives the order of the particular portion of the coefficient, and the remaining columns the parts of this order, contributed by P 1} (P 2 -r-P 3 ), (P 2 -P 3 ), respectively. Argument 1+ IQT - 18F- 2h". P 2+ P 3 PI-PS A 2 2 r" 2 -15"-89 - l"-28* + 1"-19* r" 4 + 1-65 + -17 - -09 y" 6 11 - -01 00 + -01 e'V' 2 + -27 00 01 e" 2 y" 2 + -12 + -01 - -01 e'Y 2 - -00 e'V'Y' 2 - -01 Sums - 13-96 - 1-11 + 1-08 All of these were fully calculated with the exception of that of order 7" 8 , which a brief examination showed to be between 0"'005 and 0"'015 ; the terms of orders e' 2 7 //4 > e" 2 y" 4 , and a portion from the third term of R are of about the same size, but they were not calculated. I believe the result, 13"'99, and the values for the other terms of this period, are accurate to within 0"'05 ; the additional computations necessary to obtain the final coefficient within 0"'005 are not very long, but in view of the uncertainty in the mass of Venus, which is doubtful within one per cent., and of the length of the period of the term, the present results are fully sufficient. Calculation of the Functions j3 Sip (i) . The known methods used for this purpose f have been modified in order to avoid the want of symmetry which makes them troublesome, and the considerable loss of accuracy which occurs for large values of i. The point to be considered in this connection is not the number of places of decimals but the number of significant figures, and the methods used here have been * In the original essay these portions were given to be - 1"'90, + 1"-16, respectively, owing to a numerical error. t See Tisserand, Mec. Gil., Vol. i. Chap. 17 ; and Leverrier, loc. cit. Ill] CALCULATION OF THE FUNCTIONS A (i) 29 adopted, partly to obtain the same number of significant figures for every coefficient, partly to avoid numerous multiplications by incommensurable numbers (involving the use of logarithms) and partly to obtain easy checks on the results, so that there shall be no doubt about the numerical accuracy of the tables giving the functions fi gjp (i} . It is true that functions for values of i not afterwards needed have been found, but I believe thaT the gain in accuracy more than counterbalances this defect. The sieve has shown the maximum value of i which is likely to be required in the case of each planet, and the tables have been formed up to this maximum value of i. The advantage of the present methods has been obtained mainly by the use (d \ n f d\ n a -r } in preference to a. n I -=- ) . dcLj \d + b, + b 2 + . . . = 1 + \ a? + f a 4 + . . . VI a 2 1 1 a -r\ r\ f\ n t" ----. /i - /i _ __ ri > rt - CtliU I M t L l/Q ^~ l/j l/j,^l 1 ^~ W/i . VI - a 2 Vl - a 2 1.3...(2*-1) 1 2.4...2t ' 2i+2 Then an easy transformation gives (23) A* = 2a* [(2i + 2) q i+l a - q i+l a^ - q i+2 a 2 -...]. A table of the values of a 1 was formed and also one of the logarithms of the 1), testing with i odd. For the other planets it was sufficient to find the highest values from (22) and follow down with (24) to i= 0, which was computed from (22) as a test. For the IP we have (Tisserand, loo. cit.} (25) ')(2t' + l), from which any pair of values of the B i may be computed ; there is however a loss of accuracy. But we have (Tisserand, loc. cit.) a formula which may be written (26) B { ~ 1 = ZiA* + # +1 and which enables us to calculate the B 1 with great rapidity, without any loss as soon as the two highest values of B 1 have been found ; moreover the loss in the latter disappears after the first step and the succeeding B i ~ 1 have the same degree of accuracy as the corresponding A\ For security, the B l were calculated from (25) after each ten steps, and several were tested by the formula (Tisserand, loc. cit.) 9V _ 9 9V _ 1 TV.' ^*" ** TW i ^^ * 7"i*' n * See Tisserand, Mec. Cel. , Vol. i. Chap. 17. Tisserand gives the formulae for A\ B^a, C*la?, .... t The calculations for Venus had been completed before Mars was undertaken. Ill] CALCULATION OF THE FUNCTIONS (3 Stp (i) 31 The greatest advantage of (26) however is the rapidity with which the calculations are performed by means of it, for it does not require the use of logarithms. Special values of the remaining functions C 1 , D 1 ,... were obtained from the formula (to be proved later on) (27 ) Mm ^,^^g^ff f ,' = !-,, ,=l + a , which involves no loss of accuracy, combined with (28) - /I-;W+/C. for the intermediate values. (It is interesting to notice that (27) gives 2 but no use was found for this formula.) The /8 8)P * have now to be obtained. There are two problems to be considered : the first is where a table is to be made for all values of i up to some definite place (30 in the case of Venus* and Mars, 8 for Mercury and 6 for Jupiter). We require here a formula for special values of i, and when the coefficients for the two highest values have been found, the remainder are obtained from the obvious generalization of (28): () *,, This method does not give the values for s = %', the formula for special values of i is required throughout, The second problem arises when so few values of i are needed that it is not worth while to make a full table. Hence we require to develope formulae from which the /3 g) / can be obtained for special values of i, it being understood that the tables are always formed with the assistance of (29). In the following, the index is generally omitted, since it is the same in all the formulae now to be obtained. Remembering that A - = {1 - 2a cos (T - P) + a 2 } ~l = '-<-*> 2&< cos i (T - P), we easily obtain, if for a moment we put J3 8 a~ s+ i = b s , Db s = s(l-a. 2 )b s+1 -sb 3 , Except for A, B, which were required to i = 43. 32 DEVELOPMENT OF THE DISTURBING FUNCTION [SECT, whence I fL>, = /S I I II- I L>g+l ~$PS> (30) (D-s giving (31) D 2 & = 4s 2 a& +1 + 2 ( - (32) (D + 1) 2 & = 2s ( 6 + K) & +1 By definition, / d \ P whence, if k be any number independent of a, (33) (D + k) J3,, p = (p + 1) A p+1 + (p + A) /8 8jp , (34) Again, from (30), (35) (D - * + |) 2 Combining this with (34), after putting & = s + ^, we find (36) p (p + 1) ft tip+l = 4s 2 (&+,,_, + A+i,p-2) - p (2p - The material contained in these equations is sufficient to find all the J3g tp rapidly, when the yS g are known. We have, in fact, A-iiflfc-M,, -6i = f e'C ^B 0t 2^1 2 = aB - A,+ i*A , 2B 2 = 9aC + B 1 + (i^- 2C f 2 = 25aZ> + 3^ + (i 2 - 4) (7 , 6^3 = (#1 + ^o) ~ 6^ 2 + (^ - 1) ^i 65 3 = 9a((7 1 + C' )-2 J B 2 + i 2 5 1 , 6(7 3 = 25a (A + A) + 2C 2 + (t 2 - 1) a i} . = 9a (C, + (70 - 95 3 + (i 2 - Ill] CALCULATION OF THE FUNCTIONS /9 S) p 0) are obtained from these without loss of accuracy. The functions {(D + I) 2 - i 2 } @ 8>p , (D 2 - i 2 ) @ 8 , p are required to find the P lt P 6 , P 7 ,...(page 24). If these were obtained from (34) with the values of j3 Stp just found they would in general entail a loss of accuracy of the order of i units in the last calculated figure. Equations (31), (32) enable us to calculate them without loss when p = 0. When p > 0, we have from (34), (36), {(D + i) 2 _ *j Afp = s * a (/3 s+i>p + ft^^) + (p + 1) (2s + (s + i)(2p-s + f)/3 Sj In particular, I) 2 - i 2 } .= - + 3a C - 2B , \a {(D + I) 2 - *} C = ( 15 + 10o) D. - 6(7., V a / {(D + iy - i*} B p = 9a (Ci, + C^_0 + 4 (p + 1) B p+l + 2 . {(D + I) 2 - *} C p = 25a (D p + D p _,) + Q(p + l) C p+l which are rapidly obtained, since - the multiplication by a will have been made in finding the f3 8>p . In a similar way we obtain from (31), (34), (36) for outer planets, (D*-i*)A = aB , (&-i*)B = 9aC + 25 1 - B 0) (D 2 - i 2 ) C = 25aD + 4(7, - 40 , = 25a (D,, + D^) + 4 (p + 1) Cp +1 + 2 (2p - 2) (7,. It has been found convenient to alter certain of the formulae (21) a little. We have iA* =%B i-1 - $B i+l , iB i = ^C i ~ l -f(7 i+1 . Applying the first of these to the definition of E\ the second to that for L\ and both to thab for G\ it is easy to show that (37) BR. 34 DEVELOPMENT OF THE DISTURBING FUNCTION [SECT. These, equally with (21), can be operated on by any function' of D and consequently they are true when the suffix p has been attached to the symbols. Solution of a Difficulty. From, the series for A p , or otherwise, it is obvious that (D i} q A i is of the same order of magnitude in general as A 1 . Hence when i is large and this function is calculated from the tables it appears as the difference of two nearly equal numbers, and a considerable loss of accuracy results. The same fact is true of all the functions A p , B p , ..., but for a given value of i the loss of accuracy is smaller as p increases and as we go along the series A,B, C,.... Although the number of places used in the tables has been found to be sufficient for the purposes of this essay, I shall give a method showing how the difficulty may be overcome. We have (Tisserand, loc. A i = Hence (D i)A* = Therefore ait.) : 2 |r)i " t* 7T JO (1 2 r a t+3 1 - a 2 sin 2 i/r)f 7T JO(1 a 2 sin 2 \|r)i (D - i) A i = aA i+l + = aA i+1 + a?A i+2 + ... + a.iA i+ i + & (D - i -j) A i+ J. The values of A\ DA i = Af have been calculated to seven significant figures, and the ratio of (D il)A i+1 to (D i)A { approaches the limit a as i increases. Hence if p is the number of units loss of accuracy in the last significant figure of (D i j) A i+ i, the number of units loss in (D i) A 1 is pofi approximately, which can always be reduced to unity or to a gain by taking j large enough. For values of i less than i' we must take a 2 -?' < i' if we are to lose no accuracy in (Z) i) A 1 . In the case of Venus, with i' 30, this gives j = 5, i= 25. It is obvious that this principle can be extended to any of the functions previously used by applying the preceding formulae, but since the practical applications did not demand it I shall not develope the formulae here. It does not seem possible to obtain a general formula for (D i) /3 S) / in terms of /3 S) / without loss of accuracy otherwise than by the principle of the method outlined here ; i.e., by referring forward to some higher value of i. The corresponding difficulty in Radau's work appears in the form A i - aui*- 1 , A* - Ill] CALCULATION OF THE LUNAR FACTORS 35 Calculation of the Coefficients MI and of their Derivatives with respect to n. The most arduous part of the work has consisted in the computation of the derivatives of r 2 , a? y 2 + %ixy = u 2 with respect to n, to _the degree of accuracy which was demanded. The only complete literal expansion in powers of m is that given by Delaunay, and his results suffer from two serious defects for this purpose, the one in the fact that the coefficients for the parallax on which the desired quantities chiefly depend are not taken far enough, and the other in the doubts as to the degree of convergence of the series along powers of m. Both may to a certain extent be remedied by a judicious use of the numerical results given in my Lunar Theory. So far as I know, the only method for calculating these derivatives from numerical results in which the value of m has been substituted, without having recourse to algebraical series in powers of m, is one which I gave in 1903*; this method, although tedious and difficult for numerical application, has perforce been adopted in the absence of any other plan. The method essentially consists in finding three independent relations between the coordinates and their derivatives with respect to the six elements, expressed in the form of algebraic or differential equations. From these three relations it is possible to obtain the derivatives of the coordinates with respect to one of the elements in terms of the others. No approximation processes are necessary only multiplications of series and a quadrature for each coordinate. Further, the results naturally appear in the form in which they are required in the disturbing function, namely, as derivatives of the dc rectangular coordinates, and the value of -^ , on which the principal term in the equations of variations mainly depends, also arises naturally. The order with respect to e, k to which the results can be calculated is one less than that to which the coordinates have been obtained, but it was not found necessary to proceed to high orders. The formulae are constructed to solve the problem: Given the lunar coordinates and their derivatives with respect to c 2 , c 3 , w 1} w 2 , w 3 , to find the derivatives with respect to n, it being supposed that the coordinates are expressed in terms of these six elements. My lunar theory enables one to get all the data of the problem accurately. It is expressed in terms of the constants e, k, which are left arbitrary; but the second and third of equations (8) together with the values of c 2 , c 3 in terms of these constants enable one easily to find the derivatives with respect to c 2 , c 3 from those with respect to e, k, or vice versa. The method given for performing the calcu- lations was adopted almost without change. The results will be found in the * Trans. Amer. Math. Soc. Vol. iv. pp. 234 248. 32 36 DEVELOPMENT OF THE DISTURBING FUNCTION [SECT. Ill collection of tables. The details by which the functions chiefly needed, namely, ^-W), ^(UA <..). were found directly instead of the functions for which the formulae were given, namely, d , x d . x Tn^' Tn^' will be omitted*. The greater part of the work consisted in multiplications of series of the same nature as those necessary in my lunar theory. The calculation of r*-3z\ u 2 , u z consists entirely of such multiplications. The multiplications by are quickly performed. Certain coefficients of orders higher than those given in the tables were required. These were obtained from the complete results already quoted for u , z: their derivatives with respect to n were multiplied by such small coefficients that a calculation of them was obviously un- necessary. Thus in all this work the effect of the solar perturbations in the moon's motion on the coefficients of the planetary terms has been fully taken into account. It may be stated here that the derivatives with respect to n are only required either in the terms of comparatively short period (say, a few years or less), or in those terms which do not contain the argument w^ A glance at the final form of the equations of variations will make this state- ment evident. * Some errors in the formulae of the paper referred to were found. The integrated equation (C) p. 246, requires an added constant, contrary to the statement there made : this constant may be determined by the equation numbered (14) in the paper ; and the sign before the last term of (E) p. 247 should be changed, as well as the first negative sign in equation (15), p. 244. SECTION IV. A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS. THE periodic inequalities due to the direct action of the planets are usually divided into two classes, which have received the names ' long-period ' and ' short-period.' These names are somewhat misleading in this connection, because there is no sharp line of division. I shall divide them into primary and secondary inequalities : the former are those which arise from the substi- tution of w l + 8w r for the non-periodic term of V\ the secondary inequalities will be defined as those which arise from the substitution of the variable values of the elements in the periodic terms. The majority of the primary inequalities are of long period a year or more ; but there are classes of them having sensible coefficients with periods of a month or less. Nearly all the secondary inequalities are of short period. Thus the primary inequalities in the longitude are obtained from and the secondary inequalities from ( dV iNi ^ dV x , dV * , dV = - -- 1 Swj + - Bw 2 + 8w s + -- * ^ * Bn + -= Sc - 2 s -j- -= 2 -y 3 . dw. 2 dw s dn dc 2 dc s The coefficients of the secondary inequalities will be smaller than the coefficient of the corresponding primary inequality, in general ; an examination of the final form of the equations of variations and of the values of the co- efficients in the derivatives of V shows this immediately. An exception will occasionally occur in the terms * dV * -; OC 2 , -; - OW. dc z dw z dV The largest term in -= (see page 12) has a coefficient 40 ; the coefficient in s 7 x 10 A-i 38 A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS [SECT. that in Sw l is dV Hence the ratio of the largest coefficient in -= Sc 2 to that of Sw l is dc 2 1 28_ $4 1 144 p ~ 2 la 5 ' "v ' f'y A I _, / S -&-! 10^ 1 + 2^ 1 ' A ' A s A f 2 7~io- since As 1 is never very small (it is in most cases a little greater than unity) the value of p will be large only if the two terms of the denominator are approximately equal in magnitude and opposite in sign. The sieve which follows does not contemplate this ' accidental ' approximate vanishing of the denominator, so that it would not reject the inequality in Sw, for this cause. t/ 1 / Hence the maximum value of p would appear from the sieve as the greater of the two values - 4- and l ^ S ' S 2 A, 10X, 100 ' since \ =4 + '014864 '0037444 and its least value when 44= is taken to be '01486 in the sieve. The terms in which the former is the greater are all calculated without reference to the sieve. Those in which the latter is the greater demand that s > 100" for the secondary inequality to be rejected by the sieve for 8w l when it ought to be retained. The sieve shows that such inequalities can only occur with the lunar arguments 2D 21, D - 1, and dV these have been separately examined. The term -^ -8w 2 is of the same order approximately, and the same argument applies. A similar argument applied to the terms containing 2D 2F, D F shows that the sieve would not reject the secondary inequalities in longitude as long as s < 2000", and as the upper limit for the sieve is s=3500", a secondary inequality with a maximum coefficient of 0"'02 might have been rejected by it. But the inequalities for s > 2000" were separately examined. The secondary inequalities in the latitude of terms containing 2D 2F, D F are more important than those in the longitude. A similar argument applied to them shows that the primary might be neglected by the sieve if s > 60" when the secondary should be retained. Consequently a separate* examination of these terms was also undertaken. We have then to examine the primary inequalities. Those which have periods of less than a year do not need a sieve. The terms in the moon's IV] LIMITATIONS OF THE SIEVE 39 coordinates which, in combination with the planet's coordinates can give sensible terms, are few in number and they are taken one by one according to the sizes of their coefficients with all possible terms arising from the planet's coordinates until they become insensible : the number of the moon terms thus required is quite small, as the results show, and any omissions would arise only from numerical errors. The sieve which follows_is therefore constructed for periods greater than a year. Some limitation had to be placed on the maximum period to be included ; it was taken at 3500 years, corresponding to a value of ,9 equal to 1". But values of s less than 1" which arose from the equations constituting the sieve were also included. The results show that though there will be inequalities of periods greater than 3500 years, their order is so high that it is extremely doubtful if their coefficients would be sensible. In any case the period is so long that they would scarcely be observable within historic times. It is to be noted that for very long periods it is better to expand such terms in powers of t when we wish to examine their effect on the observed position of the moon. Thus: C sin (qt + q'} = + w 3 in order to obtain k. (c) A method of constructing the period corresponding to each multiple of I, g, h to be considered, to obtain s. (d) The terms which were not rejected. These will be found with the numerical results. The only part of the actual computations which required special care was that giving the periods. It was found, however, throughout the numerical work that, the plan once formed and in working order, the size of the coefficients could be almost guessed at a glance, so that the computations were themselves of the nature of a test of accuracy, and errors tended rather to make the coefficient too large than too small. The test was gone through twice. The details which resulted in the table of coefficients retained by the sieve for accurate computation will be omitted. IV] CONSTRUCTION OF THE SIEVE 41 Construction of the Sieve. We have, as in Section II., q = (1 _ 7 "a) r " sin (V" - F) - 7" V sin (V" + V f = 2 7 " Vl - 7 //2 r" sin ( F" - h"\ Hence, putting E=^,, ^ = we obtain A -7 = " t)N, Multiplying the latter equation by %E r)N, %N + t)E successively, and making use of the equation A 2 = 2 + if + ^ in the form* we find, after a few simple operations, (39) Now the principal terms in the expansion of JV, ^ in powers of e' are a', a'e', and the equation (38) shows that when the multiple of T considered in the expansion of -^- g in cosines is not very small, -T- is small compared with (s positive). Hence we shall evidently be taking the worst case if we consider g of the orders ' 6 A 3 A 5 3 *r\AV' 3 Further, ^ dT VAV = dT VAV ~ A 1 Jf = order of -^ Jr'A 3 Aj ' * The symbol s is only used in this connection on this page, and it will not be confused with the s in the equations of variations. 42 A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS [SECT. which, for the worst case that of Venus where a" 2 = 2a' 2 approximately gives - d l a'dl The order of ^^TT^) i g approximately the same. CLJ. \A / Consider next the function !__ A 3 A 5 ' Here is of the order 2j". Suppose that these functions be expanded in powers of y /2 and then in cosines. For Venus 7 //2 < j^^ and therefore for a given multiple p of T in which p < p lt the terms factored by j" 2 will be less than those independent of y" 2 : the value of p^ which would make them greater, is seen in the course of working the method to be too large to need consideration. Hence for coefficients which do not contain \ // 2 - 10~* when 7 //2 is a factor. A I /Y /*T1 /Tf2 Second and third terms of R, order ^ -^ ^ 10~*, fourth term of R, order ^ -^ 10~*. * The parts depending on y"~ always diminish the coefficient. IV] CONSTRUCTION OF THE SIEVE 43 Next, the primary inequalities are found with where 1 m" 1 m" R = - 7 w' 2 a 2 A cos ( - -003744^, // - // /yvj nfYl f= [12-30] ^7, /' = [7-62]^. J m . m There are several cases to be considered. Case 1. i'i = i 2 = i s = 0. The second term of Bw 1 with the first term of R since the other terms of R will have the factor m 2 at least ; n d ( a 2 ' dn Coefficient in &W-L is of order 10 7 - 2 m" ( . a s m' ( 10 7>9 J.V 8 " / a' 3 \ T I coef. in -- 1 , V' 2 not a factor, V A 3 / w" / /3 \ III, I . Ui \ f . - 7 coef. m , 7 2 a factor. m \ A 3 / Case 2. ^ = 0, i 3 even. The first term of 8w l combined with the second or third terms of R ; Coefficient in Sw, is of order 109.8-fc m " / d a' 3 \ r coef. m - . s 2 m \ aT A 3 / 3. t' 2 = i'i, i\ ^ 3, i 3 even. The first term of Swj combined with the first term of R ; Coefficient in Swj is of order 10 12 ' 8 -* m" I a coef. m O /I v-'v *** . o s 2 m \ A 3 The other cases, depending on the way in which w ly w 2 , w s enter into the argument, are treated in like manner. I give next the final form of the equations used in the case of Venus. vn" 1 a' 3 a' 5 Here ^7 = 40^000' Newcomb ^bulates 24, 24^. If N 3 , N 5 are the coefficients derived from Newcomb's tables for these functions, p the multiple of T present, C the coefficient in Swj expressed in seconds of arc, we find the following expressions for Iog 10 0; the arguments are put into Delaunay's notation (il -f i'g + i"h). Additions have been made to the numbers for simplicity in the results and for parts arising from the terms in the moon coordinates containing e. 44 A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS [SECT. Argument Log C I, g, h absent 1 -5 - log s + log JY 3 , I absent, i' + t", even S re * ter {* t ~ ? ~ ? ! g * + l g V ^ *" of (3-5 - k - 2 log s + log N 3 , i' + i", odd 3-5 k-2\ogs + logN 5 , I present, i' = i" = 0, 1 6-0 - k - 2 log s + log Jf at i" even 5'0 k 2 logs + \ogp + \ogN s , i" odd 5'0 k 2 log s + log JV 6 . From the second term of R, which produces terms containing the factor , by a similar procedure I find Argument Log G = 1, i' + i" even 2-3 k 2 log s + log N 6 , = 3, 2-0 - k - 2 log s + 2 log p + log JT,, i' + i" odd 1-5 -&- 2 logs + log/) + log ^V 5 . For the other planets, similar expressions can be obtained. Except in the case of Mars as far as i = ll, no tables giving N 3 , N 5 are available. It is a' 3 a' 5 therefore necessary to construct a method for the coefficients in -^ , ^. As stated above, the parts of these functions independent of the eccentricities of the earth and planet can be expanded rapidly for a rough approximation. The terms which contain the eccentricities as factors are in general of the orders (pe') q or (pe")i (where p is the multiple of T present, q the order with respect to e ', e") compared with the order of the term with argument p (T P), as a reference to Leverrier's development shows. A margin was, however, left for the occurrence of numerical factors independent of , p, CL but depending to a small extent on q. Jupiter was the only planet in which the approximations to the coefficients came close to their correct values ; and this occurred where the approximation was comparatively simple, namely, for small values of p : a smaller margin has been necessary. The coefficients which the sieve gave as greater than 0"'01 will be found in the collection of numerical results. With a knowledge of the greatest values of N 3 , N 5 (which arise with the lowest values of p) it is a simple matter to find the maximum value of s to be considered in any given case. In fact, the maxima of s are found by equating the above expressions to 2, since the least value of C to be considered has been taken at 0"'01. Many limitations which it is not necessary to specify in detail appeared during the progress of the work ; e.g., if the multiple of I present in the part of R independent of - is 2, it is only \JL necessary to consider values of p lying between 25 and 50, and then only those in which the sum of the multiples of P, T is less than 5*. * These statements are always made with coordinates referred to axes following the mean or true place of the sun. IV] METHODS FOR FINDING k AND 6' 45 Lowest orders with reference to e, k associated with the Lunar Arguments, If any argument contains w lt w 2 , w 3 in the form* + i 3 w 3 = il + i'g + i"h, so that 1 = liy 1 = 11 T ^2> * = *1 "t" *2 ' *3 then it is easily seen from the known properties of the solution of the problem of the moon's motion, that the lowest orders are as follows : (40) | i - i' | + | i' - % (41) 2 + \i-i'\ + \i'-i' (42) 2 + (43) 4 + for i', i" even, for i', i" odd, i i + smaller of | i' i"\ for i' even, i" odd and positivef, i i' | + smaller of i 1 i" 1 | for i' odd, i" even and positivef. The first expression (40) represents the orders for all terms in longitude and parallax or in x, y which are independent of ; (41) for terms con- a taining odd powers of as a factor ; (42) for terms in latitude or in z tJy independent of -, , and (43) for those containing an odd power of as a factor. Unity has been added in the expressions (42), (43) since an odd power of 7 in R is always associated with an odd power of 7", and conversely. Method for finding the Periods to be Examined. The lunar arguments I, g, h are first considered. These are divided into classes according to the multiple of I present ; when the multiple is greater than 3, the value of p is so large that the coefficients are quickly seen to be insensible ; in fact, there are none retained by the sieve with the multiple 3. Consider any multiple i'g + i"h of g, h. All the facts needed concerning a given multiple of g, h can be inserted on a sheet of squared paper by taking i', i" as the coordinates and inserting in the squares : (1) the number of seconds in the argument ; (2) the order with respect to e, 7 of the coefficient as derived from the expressions (40), (41), (42), (43) associated with the argument ; (3) the multiple of T associated with it in the greatest term of this order in the moon portions of R ; that is, the term whose coefficient in these functions does not contain e' as a factor. The specimen given * The results were worked out with I, g, h instead of with w lt zc 2 , w s . t The positive sign is attached for convenience : the sign of the coefficient can always be arranged so that i" is positive. 46 A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS [SECT. on page 47 includes all terms of the seventh order. It is evident that only two quadrants are required. Similar charts were constructed for l+i'g + i"h, 21 + i'g + i"k, ... but that it was not necessary to insert the values of s will presently appear. For the planet arguments the table which follows was formed to give the periods of pT ^Fup to the first very long one, excluding periods greater than that of T. The last three* given are the arguments of very long periods : it is not necessary to go further, since all the periods greater than a year are either multiples of those given or are combinations of the last three with the others ; higher multiples of T, V make the difference | p p 1 \ (which gives the order of the coefficient with respect to e, e", 7") so great that the terms would obviously become insensible if s > I". We thus have all the periods of arguments independent of I, g, h. This particular table also illustrates the plan on which the sieve was worked. Arg. s \N\ log|C| For Cal. T + 3548 11-0 - -9 --5 X 12^-87 -3563 03 -3-5 4^-37 -3110 14- - -9 --5 X 9^-57 + 3096 04 no 9^-67 -2672 3 -2-4 4^-27 + 2657 2-4 -1-5 X T- V -2220 230- + -6 --5 X UT-7V + 2204 002 no 6^-47 -1782 2-3 -1-3 X 7T-47 + 1766 3 -2-2 \\T-1V -1344 03 no 2T-V + 1328 14- - -5 --5 X 3T-2V - 891 13- - -3 --5 X 10T-67 + 876 03 -2-9 8^-57 - 454 25 -1-5 X 5^-37 + 438 2-3 -7 X 13^-87 - 15 004 -2-0 X 26^-167 - 30 no 39^-247 - 45 no Formula : log C = 1*5 log s + log N 8 . N 3 Max. of s 200 80000 20 8000 2 800 2 02 002 80 8 1 -5 to be added to log | C \ when the multiples of T, V differ by less than 2. * These three are multiples of the first ^ -1145 38 1222 2405 "> 7 -5T : 4 6 r 7 -5T _4 7 -5T -954 -362 229 821 1413 2004 4 -4T 7 7 ~~* ^n 4 -4T 7 7 ~~ rp 4 -4T 7 -763 -171 420 1012 1604 \~\T 5 -31 7 41^ 5 -32' *:!* -572 20 611 1203 1795 2 -2T -382 210 2 -2T 7 801 7 -3^ 1393 6 -2T 1985 2 Q F 3 -T *Jr 7 -T 7 -191 401 992 1584 5 + T 4 07 7 592 1183 5 +^ /* ^ fjl 783 1374 7 3 T 6 2T 974 1565 7 O ^f7 1164 -2 -3 -4 48 A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS [SECT. IV For arguments containing g, h but not I, we proceed according to the order given on the preceding page (for g, h\ beginning with the lowest, which is 2. There are three terms of this order with aruments : 2h, 2h, h, and motions 801", -382", -191". Taking the first, which arises principally from the terms a? y 2 , 2xy in R in the form 2D 21, we examine by means of the second formula on page 44 all the terms which have values of s less than 3500, obtained by adding 801" to the values of s in the table given on page 46. For example, the argument 2g + 2A - WT+6V gives s = - 74. To obtain N 3 we write it 2D-21-8T+GV, where D is Delaunay's notation for the difference of the mean longitudes of the sun and moon. The value of N 3 , obtained from the term in Newcomb's table corresponding to 8T+QV, is extracted, and log C is then obtained. Before finding the terms of a given order, the maximum value of s needed for that order was found and the process was followed until the maximum value of s became less than 1". For terms containing the first multiple of I, the most convenient one of very long period is taken, say I + 16T - 18 F in the case of Venus, for which s= 13". This, after the maximum of s has been found, is combined with the motions in the table of arguments pT piV, giving a list of terms to be examined. The next first order term is 2D I, which is treated in the same manner ; the starting term is 2D - I + 8T - 12F, for which s = - 87". The same process is followed with every moon argument : it proceeds according to the orders of the coefficients with respect to e, 7, and is continued until an order is reached for which the maximum of s is less than 1". The charts and tables which were constructed, together with the expres- sions constituting the sieve, permit one to review rapidly all the possible terms without any fear that some may be omitted. All the major planets from Mercury to Neptune were examined. For the minor planets a different method of treatment would be necessary : the principal effect is probably that of a circular ring of matter at the average mean distance : but the mass would be an exceedingly doubtful quantity. It should be added that the numbers given in this section are only rough and ready approximations. Accurate values were afterwards used in the actual computations of those terms retained by the sieve. SECTION V. AUXILIARY NUMERICAL TABLES. Epoch 1850-0. Daily motions Longitudes at Epoch Arg. of arguments Perigee Node to, w., 47434-891 400-923 Mercury Venus 75 07' 19" 129 27' 34" 46 33' 12" 75 19' 47" w.. -190-772 Earth 100 21' 40" Q 14732-420 Mars 333 17' 55" 48 24' 01" V 5767-670 Jupiter 11 54' 27" 98 55' 58" T 3548-193 Saturn 90 06' 40" 112 20' 51" M J S 1888-518 299-129 120-455 Eccentricity Inclination Sine half inclin. a" log-, m' m" Moon e = ... e= ... Earth 10955 054906 016772 k -044780 y -044887 Mercury . . . Venus Mars 205604 0068446 093261 7 00' 07" 3 23' 35"-3 1 51' 02" 061066 0296063 016149 T-5878216 1-8593374 1828960 6000000 408000 3093500 Jupiter Saturn 048254 056061 1 18' 42" 2 29' 39" 011466 022 7162374 9794957 1047-35 3501-6 a diff. of masses of Earth and Moon ,=/ a sum = -0025053. BE. 50 AUXILIARY NUMERICAL TABLES [SECT. a'A p \ Venus. i p = p=l p=2 p = 3 P = 4 2-386373 1-18898 2-01958 3-56717 7-14018 1 942412 1-64375 1-97018 3-61813 7-15763 2 527578 1-49718 2-23897 3-67099 7-24792 3 323341 1-25777 2-38378 3-90485 7-38420 4 206787 1-01817 2-37596 4-18715 7-66163 5 135585 806423 2-25093 4-40529 8-06778 6 0903733 629509 2-05234 4-50184 8-52231 7 0609449 486319 1-81709 4-46256 8-93131 8 0414597 372759 1-57213 4-30024 9-21822 9 0283961 283965 1-33525 4-04108 9-33621 10 0195545 215254 1-11704 3-71519 9-26837 11 0135258 162505 922804 3-35128 9-02179 12 00939037 122264 754294 2-97337 8-62021 13 00653977 0917207 610989 2-60025 8-09624 14 00456677 0686356 491048 2-24529 7-48568 15 00319644 0512491 391967 1-91729 6-82294- 16 00224186 0381937 311003 1-62103 6-13881 17 00157519 0284159 245453 1-35848 5-45869 18 00110854 0211093 192801 1-12945 4-80230 19 000781251 0156602 150799 932354 4-18393 20 000551305 0116035 117495 764686 3-61295 21 000389493 00858810 0912272 623494 3-09465 22 000275467 00634988 0706073 505651 2-63101 23 000195012 00469065 0544896 408070 2-22155 24 000138178 00346202 0419391 327837 1-86399 25 0000979878 00255320 0322001 262283 1-55486 26 0000695400 00188159 0246666 209030 1-28999 27 0000493857 00138571 0188559 165996 1-06486 28 0000350954 00101988 0143859 131384 87493 29 0000249553 000750191 0109555 103668 71570 30 0000177550 000551514 00832894 0808952 58391 v] TABLES FOR PLANET FACTORS 51 a'Bj. Venus. i p = Q P =l p = 2 p = 3 j>=4 7-22787 40-7577 162-839 574-438 1889-97 1 6-41712 39-9376 162-009 572-558 1886-46 2 5-34305 37-4703 158-899 567-202 1875-65 3 4-30681 33-9489 153-053 557-873 1857-47 4 3-40299 29-9236 144-596 543-770 1831-35 5 2-65251 25-8036 134-045 524-376 1796-17 6 2-04714 21-8593 122-086 499-716 1750-67 7 1-56803 18-2494 109-417 470-353 1693-91 8 1-19391 15-0509 96-6472 437-240 1625-63 9 904676 12-2853 84-2627 401-549 1546-42 10 682782 9-93951 72-6128 364-502 1457-58 11 513587 7-98023 61-9221 327-246 1361-05 12 385215 6-36440 52-3111 290-774 1259-10 13 288217 5-04589 43-8189 255-885 1154-16 14 215180 3-97965 36-4253 223-167 1048-60 15 160347 3-12409 30-0696 193-016 944-563 16 119287 2-44218 24-6664 165-649 843-909 17 0886081 1-90189 20-1175 141-143 748-121 18 0657309 1-47604 16-3210 119-461 658-314 19 0487009 1-14196 13-1767 100-483 575-238 20 0360433 880955 10-5906 84-0315 499-324 21 0266486 677817 8-47687 69-8952 430-720 22 0196845 520254 6-75904 57-8446 369-349 23 0145280 398422 5-37015 47-6466 314-956 24 0107140 304484 4-25252 39-0734 267-158 25 00789548 232245 3-35707 31-9104 225-485 26 00591455 176824 2-64251 25-9593 189-415 27 00427940 134402 2-07441 21-0408 158-405 28 00314773 101996 1-62429 16-9955 131-913 29 00231406 0772887 1-26880 13-6833 109-411 30 00170032 0584848 988870 10-9828 90-4019 42 52 AUXILIARY NUMERICAL TABLES [SECT. a'C p \ Venus. i p = P =l j> = 2 44-8769 534-588 3768-39 1 43-6375 527-704 3741-20 2 40-5989 507-963 3660-39 3 36-5135 477-744 3529-34 4 31-9852 440-064 3354-28 5 27-4388 397-947 3143-75 6 23-1435 354-052 2907-47 7 19-2502 310-509 2655-40 8 15-8260 268-888 2396-86 9 12-8827 230-239 2139-95 10 10-3980 195-177 1891-28 11 8-33082 163976 1655-86 12 6-63168 136-655 1437-18 13 5-24910 113-060 1237-38 14 4-13379 92-9238 1057-42 15 3-24075 75-9167 897-406 16 2-53032 61-6831 756-725 17 1-96836 49-8668 634-298 18 1-52609 40-1283 528-726 19 1-17959 32-1543 438-447 20 909213 25-6635 361-821 21 699011 20-4082 297-239 22 536133 16-1740 243-145 23 410305 12-7778 198-107 24 313370 10-0649 160-803 25 238882 .7-90608 130-066 26 181778 6-19418 104-853 27 138097 4-84113 84-2627 28 104749 3-77495 67-5142 29 0793390 2-93721 53-9426 30 0600106 2-28070 42-9841 TABLES FOR PLANET FACTORS Venus. 53 i a'Aj 31 + -00001 26389 32 900151 , 33 641390 34 457215 35 326061 36 232619 37 166018 38 118527 39 846489 40 604735 41 432154 42 308914 43 220879 44 157975 45 113014 i a* 31 + -00172640 32 126734 33 929960 34 682120 35 500139 36 366576 37 268590 38 196732 39 144055 40 105451 41 771713 42 564604 43 412972 44 301987 45 i a'D a'Dj X3o 338-022 6140-05 4785-68 1 333-468 6086-23 _ 4743-14 2 320-567 5928-97 4618-95 3 300-989 5679-86 4422-57 4 276-750 5355-68 4167-46 5 249-813 4975-76 3869-07 6 221-872 4559-78 3542-93 7 194-267 4126-04 3203-45 8 167-972 3690-36 2862-99 9 143-625 3265-59 2531-56 10 121-595 2861-50 2216-68 11 102-033 2484-89 1923-59 12 84-9390 2140-01 1655-50 13 70-2012 1828-94 1413-96 14 57-6434 1552-09 1199-19 15 47-0520 1308-57 1010-45 16 38-1990 1096-59 846-298 17 30-8580 913-798 704-860 18 24-8142 757-498 584-005 19 19-8702 624-874 481-532 20 15-8493 513-126 395-238 21 12-5965 419-566 323-039 22 9-97758 341-697 262-975 23 7-87850 277-235 213-283 24 6-20282 224-141 172-372 25 4-87014 180-612 138-849 26 3-81398 145-080 111-495 27 2-97966 116-193 89-267 28 2-32255 92-796 71-270 29 1-80647 73-913 56-751 30 1-40221 58-724 45-076 54 AUXILIARY NUMERICAL TABLES [SECT. a' {(D + 1 ) 2 - i 2 } Aj. Venus. i p = P = l p = 2 p = 3 9-99247 46-3548 178-7689 615-388 1 8-87162 46-3419 177-6337 613-923 2 7-38673 44-4156 175-2606 608-892 3 5-95414 40-9800 170-6122 600-643 4 4-70461 36-6645 163-2380 588-520 5 3-66708 32-0061 153-3097 571-636 6 2-830 If) 27-3902 141-3932 549-461 7 2-16779 23-0619 128-2057 522-054 8 1-65057 19-1572 114-4568 490-025 9 1-25071 15-7336 100-7589 454-380 10 943940 12-7974 87-5890 416-333 11 710030 10-3226 75-2842 377-131 12 532556 8-26618 64-0535 337-940 13 398458 6-57744 54-0018 299-757 14 297484 5-20435 45-1531 263-374 15 221679 4-09735 37-4736 229-369 16 164913 3-21139 30-8897 198-119 17 122500 2-50685 25-3054 169-824 18 0908724 1-94975 20-6139 144-540 19 0673285 1-51142 16-7052 122-211 20 0498295 1-16808 13-4733 102-699 21 0368414 900236 10-8190 85-8106 22 0272136 692034 8-65228 71-3175 23 0200849 530730 6-89346 58-9776 24 0148120 406135 5-47294 48-5458 25 0109154 310161 4-33096 39-7849 26 00803856 236420 3-41683 32-4717 27 00591623 179894 2-68796 26-4008 28 00435171 136657 2-10891 21-3874 29 00319917 103652 1-65045 17-2666 30 00235068 078504 1-28460 13-8970 v] TABLES FOR PLANET FACTORS 55 a! {(D + I) 2 - (i + I) 2 } B p \ i positive. Venus. i p = p=rl P = 2 447-951 5197-28 36045-4 1 424-582 5055-43 35471-2 2 387-464 4804-84 34419-0 3 343-350 4470-38 32933-5 4 297-290 4080-18 31081-8 5 252-663 3660-99 28946-7 6 211-488 3235-50 26618-6 7 174-796 2821-32 24186-4 8 142-934 2431-00 21731-0 9 115-818 2072-54 19321-6 10 93-1103 1750-22 17012-9 11 74-3419 1465-46 14845-2 12 58-9993 1217-63 12845-7 13 46-5732 1004-69 11029-4 14 36-5893 823-755 9401-82 15 28-6228 671-520 7960-90 16 22-3046 544-536 6698-96 17 17-3203 439-428 5604-44 18 13-4070 353-028 4663-47 19 10-3476 282-448 3860-91 20 7-96501 225-117 3181-42 21 6-11593 178-788 2609-92 22 4-68543 141-525 2132-23 23 3-58195 111-684 1735-17 24 2-73298 87-8809 1406-89 29 2-08143 68-9647 1136-77 26 1-58250 53-9832 915-517 27 1-20125 42-1554 735-058 28 910479 32-8454 588-447 29 689121 25-5372 469-778 30 520888 19-8153 374-056 56 AUXILIARY NUMERICAL TABLES [SECT. a {(D + I) 2 - (i + I) 2 } Bj, i negative. Venus. i p = j=l p = 2 447-951 5197-28 36045-4 1 450-251 5215-18 36119-2 o 430-208 5104-60 35690-2 3 395-032 4877-77 34770-1 4 351-738 4558-96 33395-3 5 305-713 4177-06 31627-6 6 260-619 3760-12 29548-7 7 218-701 3332-30 27250-1 8 181-139 2912-63 24823-7 9 148-386 2514-81 22355-1 10 120-4216 2147-80 19917-4 11 96-9397 1816-59 17569-8 12 77-4986 1523-12 15356-6 13 61-5605 1267-08 13308-0 14 48-6395 1046-62 11441-6 15 38-2436 858-965 9765-07 16 29-9390 700-836 8277-61 17 23-3457 568-756 6972-43 18 18-1396 459-303 5838-58 19 14-0489 369-237 4862-34 20 10-8485 295-593 4028-67 21 8-35441 235-725 3321 98 22 6-41766 187-307 2727-03 23 4-91853 148-339 2229-22 24 3-76152 117-111 1815-13 25 2-87098 92-1892 1472-47 26 2-18721 72-3729 1190-34 27 1-66343 56-6708 959-094 28 1-26303 44-2685 770-368 29 957552 34-5027 616-958 30 724926 26-8335 492-720 TABLES FOR PLANET FACTORS 57 a {(D + 1 ) 2 - i 2 } C p \ Venus. i p = j=l 9185-44 163970- 1 9065-52 162567- 2 8722-88 158461- . 3 8199-80 151939- 4 7548-99 143425- 5 6822-81 133414- 6 6067-08 122420- 7 5318-32 110922- 8 4603-33 99340-4 9 3940-00 88020-2 10 3338-69 77225-0 11 2803-95 67142-4 12 2336-00 57890-6 13 1932-08 49530-5 1-1 1587-52 42077-2 15 1296-63 35510-7 16 1053-26 29786-5 17 851-310 24843-7 18 684-906 20611-9 19 548-705 17016-9 20 437-852 13984-5 21 348-138 11443-0 22 275-855 9325-72 23 217-905 7571-41 24 171-617 6125-22 25 134-788 4938-65 26 105-589 3969-32 27 82-5145 3180-71 28 64-3347 2541-55 29 50-0521 2025-38 30 38-8608 1609-93 58 AUXILIARY NUMERICAL TABLES [SECT. Jupiter. i 4.< a"Af a"Aj af'Af 2-018865 0385392 -0209301 00174900 1 194930 200512 -0086394 00333975 2 0281439 0571840 -0304292 00192985 3 00451137 0136850 -0140702 00522212 4 000759075 00306243 -00467341 00325231 5 000131349 000661353 -00133908 00137257 6 0000231472 000139706 i 418302 490177 -116415 0540404 1 118939 255013 -163696 0402207 2 0284421 0891538 -0991362 0473610 3 00636279 0262769 -0419796 0325014 4 00137387 00704369 -0147155 0160281 5 000290196 00177743 -00459230 00648290 6 0000603839 000430147 -00132477 00230235 i w " * 768937 3-01626 5-98351 619377 1 586193 2-64675 5-68166 535436 2 362211 1-93221 4-76074 384900 3 200324 1-24869 3-56256 245667 4 103463 741711 2-43259 144511 5 0510119 414433 1-54498 0801264 6 0243209 221116 926391 0424868 7 0113048 113801 530331 0217562 8 00515116 0569059 292314 0108334 VALUES OF MOON FACTORS 61 Values of Mi and of their derivatives. Arguments in the Moon portions of R. Terms containing il' in the argument are to be multiplied by e' 1 ' 1 . d dn For brevity, -^ -y- (J^a 2 ) has been printed n -^M i . Argument r 21' 3r M, + -99276 + 1-0005 + 1-259 +1-65 M 2 -0140 x 2 0835 - -201 M, 0- + -0107 + -040 4^ -1-3240 - 1-3478 -1-701 -2-21 d . n -f- M , + -0540 x 2 + -3245 + -852 +2-0 cfo rf J/ 0- + -0509 -302 0- ^M, + -0842 + -1009 + -141 de d M + -0558 x 2 + -2474 + -377 de 2 d o- + -0368 -029 d 5390 d M + -0163 x 2 dk 2 Argument i *c + n *(,-n *( + o * 9 ',m \H, 49080 - -3452 - -6736 -370 - -978 1*. 2862 - -421 799 - -73 -MO e -"' + -3108 + -477 + -889 + -83 + 1-34 n d , f + -5926 + -1813 + 1-1544 + -065 + 1-904 e dn - d - J/, + -7523 + 1-1117 + 2-5564 +2-151 + 3-595 e dn * 2 *>? /T /If ~ ~I ^fct ! e ~T~ -i 6" dft - -2228 - -663 - -400 n d - -y- J/. 2 e~ on - -8064 -1-81 93 " .17. e 2 C/74 -6010 d , e -j- A de -93 -27 Argument 1 2F + 2-9901 (2F + V) + 2-8805 (2F - 1') + 3-0882 1 k" 2 M * - -2001 - -25 - -37 F^a + -1717 + -19 + -27 k 2 dw J/1 - 2-9549 -2-937 - 2-939 r. ^ 2 1C" CvTt + -3785 + -509 + -645 n d i > "i s k'- an -2873 -275 + -349 k ^ J ^ = 2Jf Argument 1 k- ! (2F - 2D) - -2062 -8051 (2F - 2D - I') -1653 1 F - + 1-9756 + 6-22 -2-31 & + 1-9690 + 6-18 + 2-31 ti d k- dn + -4637 + 1-967 319 k 2 ^4 - -1-9243 -6-282 + 2-583 k^^ + 1-9019 + 6-164 + 2-605 64 Argument AUXILIARY NUMERICAL TABLES + 1-0000 + -8917 +1-1175 [SECT. -1-1646 -1-0207 -1-3401 3E Argument h-h" h-h" + l' h-h"- I' M * -1-0004 -1-0838 9285 1 T- M * k dn + 1-16399 + 1-3385 )> + 1-0256 Argument IM< 2T-h-h" 0457 2T-h-h" + l' 1887 2T- h-h" -I' - -0041 + -1213 + -7669 - -0592 Argument 2D + h - h" 2D + h - h" + 1' 2D + h-h" - I' J M, + -0355 + -1421 - -0078 k , - M. - -0857 - -3682 + -0227 k dn Argument D 4D-3Z 4D-4F /-2F 31-2D M 1 + 1132) 002258) -00 ) -3-497 + -02080^ M + M + -7182[ a, 04854 1 e 3 - -0334}- k^ - -3876 - ek 2 + -04856 [ e 3 M]-M[ --0180J 00066 J --0004J + -9779! --07420J Argument 4D - 1 - 2F D - 21 3D - 2F 1 - D M l + 01108) -1387) + -0951) + -00396) M + M. + -02282 [ ek 2 + -048 [ eX + 1-960 [ k 2 ai - -04627 (- ea, M[-M, - -00056 ) + 310 ) -006 ) - -58098.) Argument l + 2g + h-h" l + h-h" -2l-2g + h-h" 3g + ih-h"-2T M, -l-49ek + -49ek - -989k 3 - -0154e 3 k Argument D D-l' D + l' l-D l + l'-D M 6 + M 7 + 1-972 + 5-143 \ +'815 i - -46201 -1 ' 38 1 M M - -0280 134 , --056 , -2-465 -5-68 D / > e V e ye y ee M 6 + M 9 - -0139 058 1 - -389 1 + -0051 f + -023 | M 8 -M 9 - -000 + 0009; +-0005J + -0227J + -089j Argument 3D-2F 3D - 2F - I' M 6 + M 7 326^ + -62"| M e -M 7 + ^Hu ' 1 M* 8 -Ml + 1-959 1 k oooJ - 3-48 f k ' e o J v] EXAMPLE OF THE COMPUTATION OF A TERM 65 Extract from the computations for illustration of the method. = (/) + (297* - 26 F-3ar') Leverrier (270)* = (112), i = 26 ; K i = A\ J^ tor e (%J ' / p'\ 3 Lev. coef. [^ (27 + 65i + 42i 2 + 8i 3 ) 4 * + ^(17 + 17i + 4i 2 ) A^ + (5 + 2i P= 28453^ + 1581-S^j + 57A 2 +A 3 for i- 26 DP = 30034-5^ + 3277^2 + 174 A 3 + 4^ 4 [DK V = From the tables for Venus P= + 6'568, DP= + 178-87, {(/>+ l) 2 -t 2 } P = 829-8 - /e'\ 3 Factor ( ) P 3 = 26 (D + 2) />= + 4992, P^=\ P 1 + (D + 2)P+(2V-1)P = + 5041 27-85 1 log log 1 Sw^ !/ r *^ i y i * ^ PT1 p * ui p, + 3-1104 -2-5552 r 6' P l + 2-9190 l(P^ + P 3 } +3-7003 KA-A ) +1-3892 -0063 6-6829 1 -OSQfi /' 2-0068 M, -1-6909 fac. - 4-6099 J/ 2 - J/g +2 -3909 J/a + J/" 3 fac. ' -4-6099 fac. -1-7760 -4-6099 prod. i u o y D + T-2198 + 0"-166 -2-7011 -0"-050 + 3-7751 + 0"-006 -2-8392 - 1-6016 (1) 7-7707 insen. x fac. -4-6099 Portion in this term depending on All found from the tables except _ , y , Replace A p l by - y" 2 EJ = - ( p The three parts are which is only needed within 50/ o of its true value. It can be computed from the general formulae, but the following estimate gives it with sufficient accuracy : [(Z>+l) 2 -26 2 ]^ rt 26 1-8724 - 53270 E l 62-82 - 99360 E 2 1047-9 - 59730 # :s (est.) 12000- 12000 P -5-3502 M -1-6909 fac. - 4-6099 y" 2 4-9429 -5-7466 + 2-3909 -4-6099 4-9429 - 4-0055 -1-7760 -4-6099 4-9429 -2-5939 -0"-039 + 3-6903 + 0"-005 -3-3343 -0"-002 - 224000 etc. BR. 66 AUXILIARY NUMERICAL TABLES [SECT o'\5 + C where G depends on h", vr' ', &", and on the coefficients. Each term is set down. The single term, argument <, which is obtained by combining them, is then given. The secondary terms, arguments < + $', where ' consists of the lunar arguments only, follow. Secondary terms having the same arguments as certain primary terms have not been added to the latter but are given separately in their proper places. (See Addendum.) COEFFICIENTS OF SINES IN LONGITUDE. Venus. Short Period Primaries. i l 0(2D-Z) 9f>2Z) 0*2* 1 + 0"-478 + 0"-076 - 0"-005 + 0"-006 + 0"-005 2 + 195 + 33 + 2 + 2 3 + 102 + 19 4 4- 59 + 12 5 + 35 + 8 6 + 22 + 5 7 + 14 + 4 8 + 9 + 2 9 + 6 10 + 5 11 + 3 12 + 2 SECT. Vl] VENUS TERMS IN LONGITUDE 71 / 0Z -3 -0"-002 _ 2 5 -1 10 20 - 0"-003 1 65 11 2 + 102 + 19 3 + 28 + 6 4 + 15 + 2 5 + 10 + 2 6 + 7 7 + 5 8 + 3 9 + 2 10 + 2 i 0? -1 + 0"O01 + !_ 1 + 7 2 + 20 + 0"-004 3 32 7 4 9 5 5 6 3 7 2 - F) i l 1 - 0"-002 2 3 3 15 - 0"-002 4 + 3 5 + 1 i

= 2D-l + i(T- V) [SECT. i -17 - 0"-002 -16 3 -15 3 -14 4 -13 5 -12 7 -11 9 -10 10 - 9 12 - 8 15 - 7 17 - 6 20 - 5 25 4 28 - 3 32 - 2 32 1 30 23 1 12 2 7 3 5 4 3 5 - 2 i -6 - 0"-001 5 1 - 4 1 -3 2 -2 2 -1 2 1 i * + l + 1 - 2D + 2l -17 + 0"-002 -16 + 3 -15 + 4 -14 + 6 -13 4- 8 - 0"-002 -12 + 10 2 -11 + 0"-002 + 13 3 - 10 + 3 4- 17 3 - 9 + 3 + 22 4 - 8 4- 4 + 27 5 + 0"-002 - 7 + 5 + 32 6 + 2 - 6 + 6 + 39 8 + 2 5 + 7 + 46 9 + 3 4 + 8 + 53 11 + 3 3 + 10 + 58 12 + 4 - 2 + 11 + 60 12 + 4 1 + 11 + 57 11 + 4 + 9 + 46 * + 3 1 + 5 + 26 5 + 2 2 + 4 + 17 3 3 + 3 + 12 - 2 4 + 2 + 9 2 5 + 7 1 6 + 5 1 ^ + 4 8 + 3 9 + 2 10 + 1 The argument vanishes identically and the term disappears. VI] VENUS TERMS IN LONGITUDE i < $-1 0--/ + 2D -14 + 0"-001 -13 + 2 -12 + 3 -11 + 3 -10 + 4 - 9 + 5 - 0"-001 - 8 + 6 1 - 7 + 8 - 2 - 6 + 10 2 - 5 + 0"-002 + 12 - 2 4 + 3 + 15 3 - 3 + 3 + 18 4 - 2 + 4 + 20 4 1 + 5 + 21 4 + 5t * 4 1 + 3 + 17 3 2 + 3 + 13 3 3 + 2 + 11 _ 2 4 + 9 _ 2 5 + 7 6 + 6 7 + 5 8 + 4 9 + 3 10 + 2 i ;_!_ ,-, + 0"-001 1 + 2 2 3 + 2 3 03 a 4 1 5 4 CO + 1 5 + 1 6 + 1 - F) * Argument is zero. t Argument is I and this coefficient therefore pro- duces a slight change in the value of e, which is however too small to affect any term sensibly. i * ,- -1 + 0"-001 + 1 1 + 2 2 + 3 3 + 3 4 0) 3 + 3 5 'S3 a a> + 3 6 .9 + 2 7 + 2 8 + 2 9 + 2 10 + 1 11 + 1 74 INEQUALITIES DUE TO DIRECT ACTION [SECT. - V) i + / &+1-2D -1 + 0"-002 + 4 1 .+ 1 + 7 - 0"-001 2 + 1 + 8 2 3 + 2 + 9 2 4 + 1 + 10 2 5 + 1 + 9 2 6 + 1 + 9 2 7 + 1 + 9 _ 2 8 + 1 + 7 1 9 + 6 1 10 + 6 1 11 + 4 12 + 3 13 + 3 14 + 2 15 + 2 i * ,- -13 -0"-001 -12 1 -11 _ 2 -10 2 - 9 3 - 8 3 - 7 J 4 - 6 3 '55 4 g - 5 V t 5 c - 4 6 - 3 6 - 2 G 1 6 5 1 2 2 1 - V) i * , +> 2 - 0"-002 3 _ 2 4 3 5 p2 3 6 '35 3 7 1 3 8 3 9 2 10 _ 2 11 2 VI] VENUS TERMS IN LONGITUDE 75 i + l -l + 2D 0-2Z -2l + 2D -5 - 0"-002 -4 3 -3 4 2 7 -1 13 + 0"-003 - 0"'002 30 t - 0"-002 1 4 58 + 12 4 2 11 140 + 28 10 + 0"-003 3 76* + 722 - 0"-007 147 + 49 13 4 + 3 + 95 19 + 6 2 5 + 46 9 + 3 G + 28 6 + 2 7 + 18 4 8 + 12 2 9 + 9 10 + 6 11 + 4 12 + 3 13 + 2 14 + 2 * s= -363 '9, period =9J years. t See second note on p. 73. + 2D -2Z) -l-2D 0-Z+Z' ^ -- 1 - r -l-l' + 2D - 21 + 4D $-31 3 -0"-010 + 0"-008 + 0"-007 + 0"-005 - 0"-004 - 0"-007 - 0"-002 + 0"-003 76 INEQUALITIES DUE TO DIRECT ACTION [SECT. i -l 0-/+2D $-21 1 -0"-003 2 6 3 12 + 0"-002 4 - 0"-005 42 + 9 - 0"-003 5 + 34 7 + 2 6 + 12 2 7 + 7 8 + 5 9 + 4 10 + 2 l" + i(T- F) i * >: ,-,+ 3 + 0"-002 4 c + 4 5 'to + 13 - 0"-003 Ej 6 Z 10 + 2 7 4 8 2 - F) j * ,., i ,-, ins. - 0"-002 ins. + 0"-004 -1 5 See long period terms. > = /?, f {(i + 2)2 7 -iF See long period terms. VI] VENUS TERMS IN LONGITUDE 77 Venus. Long Period Primaries. 0o = 8T-5K, s = -452"-9 =13T-&V, 6-=-14"'85 period 7 '8 years period 239 years c -s/-2&" + 0"-007 -ro'-4/i" + 0"-0028 w" 2/i," 2 0o m" 4A" 9 00 2t<7 Iff 00 .SOT ZZ7 an, 30 +112 = +0"-OOT -z8r'-2c7"-2/i" + 10 0o ~~ 3w 2/i 1 -39 = + 0"-003 period 1920 years - OT '-6A" + 0"-34 + ? + 0"-019 -OT"-6A" 5 0(/-2Z>) + 4 0o - 3s/ - 4A" +7 + 2Z> +4 G>i, OS7 Zi/t, + 33 = + 0"-35 f = _;_ 16T+18F, 5 = period 273 years - 2/i" - 13"-99 + ? -0"-766 0+(2Z>-/-/' ) -0"-008 - 2s/ 66 0-? 815 i . / //

-l + f 6 00 1w" 20 -2D + l 168 J. j. O fp U> T -i/' + 29 0o - or' + w" - 2/t" 8 0(2Z> + ?) 20 + 3? 4 00 + Iff Iff "l 27 0+ 2Z> 166 + (4/>-2?) - 2 0o + 2m' - 4/t" 6 G) _ ix 168 + (4Z)-?) 4 0o + sr' + sj" 4/t" + ' 5 -?') 11 0Z> + 5 + (Z + F) + 4 78 INEQUALITIES DUE TO DIRECT ACTION [SECT. = I + 29T - 26 V, s=- 27"-85 0o 0o 0o 00 period 127 years as 1 - 2A" + 0"-096 as" - 2A" 25 3ra-' + 79 cy f ___ ft i -I t _^ 9__" | )) ?>TS" - 2 3 1 1 2tzr' + TO-" 2A" + rar' - 2or" - 2A" - 473-' + Iff" 0"-108 + 0"-006 period 8 '4 years + 0"-030 4-1 + 0"-005 4 + 1 + 0"-002 22 (T- F), a = -1795" period 2'0 years + 0"-002 T- 23 F, 6- = - 465"-8 period 7 '6 years - -at + 0"-009 0o CT" 3 -88=0 +0"-006 -V), 6-=788"-8 period 3 '5 years +0"-011 + 1 + 0"-003 = 21) -I + 19 (T- F), * = period 2-5 years + /; -002 (2V, s = + 87"-07 period 41 years -4A" -0"-032 J O ' Ol," O

2h" 0"-036 insensible < -2sr' 56 -l + 0"'004 ^ ' -_" i Q~ Qjii Zu ~~" tU i Of./ _i O " P\ (7).. ^ Zo tJ ^> + 1730'=<^ +0"-065 period 1-7 years insensible ^ _ 2Z> + 0"-002 I +86

+ 2D-l 16 <{>-2l + 6 J.P 2D + 3 (T 7 F), s = 819 "5 period 4*3 years 4> - 0"-002 $-1 -0"-003 = 2F- 2D + QT- 5 F, s = - 71"'27 period 50 years ^ o _ OT ' +0"-071 // .) i -2F +0"-002 M - W + 24 (T- V\ s = 61"-07 period 58 years ^ +0"-003 -2F - 0"-003 < f ) = D-l- 2F+ 15 (T- F), ,9=-30"-7 period 116 years < - 0"-002 fn S T+-2V + h" -2848" 1 T+3V + h" - 628"-7 5 T+V + h" + 1590-8 2 T+5V+h" + 3810- period 40 years < 4- t + OT" A" + 2 period coef. primaiy secondary 1 years + 0"'005 2 years + 0"-016 <(>-2F + 0"-003 2 years - 0"-009 9 years - 0"-003 4> = l+h+lGT-I8V+h", s=-203"-8 period 17 years $ + 0"-008 +0"-002 80 [SECT. period 28 years + 0"-003 period 50000 years < + o"-02 -- - D - 12T + 15V, = 5< period 71 years 0o - TO' - 2A" - 0"-008 5 + 0"-013 0,(T-F), s = -502"- period 7 - l years + 0"-002 period 54 years - 2A" 0"-006 12 6 +0"-013 period 560 years - w ' + Q"-003 - w" - 0"-001 - 88 = 0"-002 Venus Mercury Jupiter Mars -0"-0152 1 + 24 + 3 period 94 years 0o -/ 0o-^" + TO-' 2A \ 18[ of fac- 93 " if, , , a tored by + 0"-044 00 Iff' 00 W 00 93 = + I + 2 - 21 + 0"O05, -0"-002' from first 7^55-4 I part of A +0"-040 +0"-029 + 0"-002 - 0"-005 + 0"-002 * This period is approximate only. But the coefficient is insensible to observation whatever be the period. VI] JUPITER TERMS IN LONGITUDE Jupiter. Short Period Primaries. 81 i * $1 1 - 0"-069 -0"-008 2 - U"-001 - 0"-005 i 1 - 0"-002 2 22 3 10 4 3 i -l 0-Z+2D - 0"-002 -1 4 -2 - 0"-007 39 + 0"-008 -3 3 17 + 3 -4 5 i * ,- -3 insensible + 0"-006 -2 See long pen od primaries i -i - 1 + 0"-003 * 1 + 0"-003 2 + 0"-002 + 7 3 + 1 + 3 The argument is zero. i t-l 0-Z+2D *- - 0"-003 -1 insensible 7 + 0"-001 -2 See long period primaries -3 + 23 -0"-005 + 0" -4 insensible + 4 i * *-, -2 -3 - 0"-001 -0"-005 - 0"-002 BR. 82 INEQUALITIES DUE TO DIRECT ACTION [SECT. Jupiter. Long Period Primaries. = J, s = 299"-13 = 2J, s = 598"-3 period 11 '8 years period 5 '9 years 4 - or" - 0"-208 ^ -w' +0"-021 < + 173 50' = ^ +0"-209 4 + 1 4+2D + 0"-021 + 0"-002

= 21 - 2D - 2 (T - J), s = - 203"-58 period 15*5 years 4 - 0"-187 4o = 21 - 2D + 3J - 2T, s = 95"-55 period 37 years -0"-190 CD v + 0"-811 CDf) "^ ZST + 0"-017 CD ~l~ ' 7 i i 1 ^7Q 1C' J. CpQ +1(O 1O

170 <^-^ + 0"-306 4-1-2D + 7 <# + ^ + 5 4-21 + 56 <^>-Z + 2Z) 58 4-21 + 2D 15 4-1-2D + 3 4-1-1' 4 4-21 + 20 t 7 /' i ) CD ~"~ t* v T ^ j D - 8 4-2D + 6 <^> i + r f 5 4 + 2D 2 4-31 + 4 4-21 + 2D 6 4+2D 13 4-l-l' + 2D 3 4-2D + 9 4-1 + 1' + 2 4-21 + 4Z> 2 4-l-l'+2D 3 4-1 + 1' 4 ", s=-502"-6 period 7 years insensible -2T-2^", s=394"7 period 9 years 0"'002 $-1 -0"-007 (T-J\ s = -979"-8 period 3-6 years 4 + 0"-002 4 - 1 - 0"-009 4-1+21) +0"-002 period 33 years 4 - 0"-004 VI] MARS TERMS IN LONGITUDE Mars. Short Period Primaries. 83 i 4> l 1 -0"-019 - 0"-002 2 8 3 3 4 2 i 00 ~ ro '" o -"&' -1 - 0"-002 5 1 55 + 0"-008 See long period 2 + 9 1 primaries 3 + 4 i -2w" 0Q - TO 7ZT = 00 + 63 1 - 0"-002 -0"-002 2 10 + 0"-003 12 3 + 4 1 + 5 4 + 2 + 2 5 + 1 + 1 i 3 -0"-002 4 + 1 i 0-Z ^ - Z + 2D 2 -0"-002 3 5 4 - 0"-002 + 17 - 0"-003 5 + 3 6 + 1 i 00 ~ w " 0o~ 'Z* 7 ' + 211-i -4 -0"-003 -5 -0"-014 + 0"-002 See long period primaries -6 + 0"-003 62 84 INEQUALITIES DUE TO DIRECT ACTION [SECT. Mars. Long Period Primaries. a = 2M-T,s = 224"-8 fa = 21 - W + QM - 5T, s = - 127"'31 period 16 years period 28 years ^o-nr" -0"-055 < -w" -0"-OH < n tzr' +8 < n t<7 + 213 = < + 0"-060 4> + 211 = -l + '2D + 8 <-2 3 21 - W + 8M - 6T, s = + 97"'56 period 38 years 4> -2OT" -0"-015 w' nr" + 5 -0"-019 -0"-030 + 6 2 4D - 3Z + 25Jtf - 23T, s = - 0"'58 =h period 6000 years period 104 years -2vr" +0"'03 ^-w"-h" -0"'015 < ra' in" 1 -0"-04 ^>,, + 165 = ^ +0' / -017 I - 0"-002 Mercury. No Short Period Primaries. Long Period Primaries. % period 39 years period 7 -9 years -2 OT " -0"-047 4>o-^" -0"-004 i / n q j ' , 1 40 +0"-003 75 = ^> -t-0"-075 T~ + 0"-006 I + 0"-004 VI] VENUS TERMS IN LATITUDE 85 b = - 2F + 1 - 3r + 4>Q, s = 67"-75 = l + 2g + 3h-T- 3Q+h", s=-100"-41 period 52 years period 35 years -0"-004 + 0"-002 0"-003 Short Period Primaries. = 2D-l-2(S-T) Saturn. (f> + I - 0"-001 -S\ s=-560"'9 period 6 '3 years < insensible $-1 +0"-014 -l+2D 3 Long Period Primaries. <> a = S, s = 120"-45 insensible + 0"-003 period a 30 years -0"-026 (S-T) $0-' + 2 <(> + 90 = + 0"-024 0"-003 period 8 years insensible - 1 + 0"'004 COEFFICIENTS OF SINES IN LATITUDE. Venus. Short Period Primaries. i -F 4 + 0"-002 5 See long period primaries 6 -0"-002 1 < + F 7 +0"-002 6+3 5 + 5 4+6 3+9 2 + 14 1 +27 See long period primaries 1 15 2 6 3 3 Long Period Primaries. period 9| years F -0"-003 4>F-l + 32 + F-l + 2D 6 <}>-F-l+2D 5 -F-2l + 4 2^-2D+6T-5F-90 , 5=-7l // -27 period 50 years -F +0"-068 -F+2D 2 $-F-l + 4 + ~ F + l 4> = h- h", s = - 190"-8 period 18-6 years / , s=74"-06 period 48 years +0"-003 F-l + 4 period 41 years -F -0"-044 -F-l 2 -F+l + 2 = h + k" - 5T + 3V, s = - 628"-72 -^j 5 . 6 years F-l 13 13 VI] JUPITER TERMS IN LATITUDE 87 ,s= + l"-85 =-Z-16r+18F-15059', s = period 1920 years <(>F + 0"-016 $ = l + 297- 267+ 112, s = - 27"'85 period 127 years F +0"-005 <=2D-Z+2lT-20F-870',s=-101"-92 period 35 years F + 0"-006 period 273 years F - 0"-650 (F-2D) + 22 (F+2D) 12 + F+l 79 - F-l 56 <(> + F-l 7 Q-F+l + 14 + F-1 + 2D 13 -F+l-2D 17 ^ + +F + -2l L? 6 + (F-I-2D) - 3 Jupiter. No Short Period Primaries. Long Period Primaries. _ r // _ OQO/'.I (D U ~ Ttf j &iJU X period 12 years 4, + ^ +0"-002 period 37 years F + 0"-009 -lF + U 2 3 ->v" > s= 108"-36 period 33 years < + F - 0"-005 > = 2F-W-2(T-J),s = 979"-8 period 3-6 years -F - 0"-026 = 2^- 2D - 3 (T - J), s=- 2269" period 1-6 years t-F +0"-005 <#> = 21 - 2D - 2 (T- J), s = - 203"-58 period 15-5 years F -0"-008 ^^F-l + 35 J i V 7 3A ro -J- -/* t- OD if / i_ o r) n (O ^ X ~* C< T~ iM--* vl j i Z?^ 7 i O 7^ 7 Q) ~r~ v \ &U i -F-2l + 4 O7 9 ^6 -f- ^j = h-h", s = -190"-8 period 19 years + F + 0"-038 = h - h" + T-J,s = 3058" period 1-2 years + F - 0"-002 88 INEQUALITIES DUE TO DIRECT ACTION [SECT. VI Mars. (f> = h + 2M-r+ 165, s = 34"-07 =h- k", s = - 190" '-8 period 104 years period 19 years + F -0"-010 $ + F + 0"-004 Mercury and Saturn. None. COEFFICIENTS OF COSINES IN PARALLAX. <(> = 2l-2J) + 3(T- V) <}>-l +0"-006 l +0"-007 $-1 +0"-007 173 <>-l + 0"-003 ADDITIONS TO ANNUAL MEAN MOTIONS. Perigee + 2" - 69 Node - 1"-42 ADDENDUM. FINAL VALUES. The primary coefficients in longitude having short periods can be combined with the secondary coefficients having the same arguments in many cases. These combinations have been made in the following tables which Therefore represent the final coefficients due to the direct action of the planets, with the following exceptions : Venus. All long period primaries together with their secondaries except those arising from the arguments 2l-2T> + 2T + i(T- V) + a, 2F~ 2D + 3 (T- V), 2F- 2D + 6^-57- 90, which have been included. Mars. The long period primaries with arguments 4D-3/ + 25J/- 23^-1 13, h + 2M- T + 165. Saturn and Mercury. All terms. These terms, omitted here, can be taken immediately from the tables in Sect. vi. Final coefficients of sines ofi(T V) + arguments at top of each column. 27 7 +26-6 2D 20-^+78 i - 4 + 0"-001 - 15 + 0"-001 - 15 + 0"-001 1 +0"-480 -3+ 2 -14+ 2 -14+ 1 2 + -200 -2 + 2 -13 + 2 -13 + 2 3 + 92 -1 + 3 -12 + 2 -12 + 2 4 + 60 + 4 -11 + 3 -11 + 3 5 + 38 1 + 6 -10 + 5 -10 + 4 6 + 25 2 + 8 - 9 + 6 - 9 + 4 7 + 17 3 + 37 - 8 + 8 - 8 + 4 8 + 12 4 8 - 7 + 8 - 7 + 5 9 + 8 5 3 - 6 + 11 - 6 + 5 10 + 6 6 - 4 - 5 + 11 - 5 + 7 11 + 4 7 - 1 - 4 + 10 - 4 + 7 12 + 1 - 3 - 36 - 3 + 4 2 + 26 - 2 + 4 T -87-8 - 1 + 15 1 + 3 -3 - 0"-001 + 16 + 2 2 4 1 + 15 1 + 1 -1 8 2 + 8 15 3 + 4 2D- 2^-18 1 47 4 + 4 -6 - 0"-006 2 + 76 5 + 4 3 + 21 6 + 3 4 4 12 7 + 3 5 + 7 8 + 3 6 + 6 9 + 2 7 + 4 10 + 1 8 + 1 90 ADDENDUM. FINAL VALUES 1 2D -l-T+89 21 8 - 0"-002 2 + 0"-001 - 10 - 0"' 001 -3 -0"-001 7 4 3 + 6 - 9 2 2 - 2 6 5 4 1 - 8 - 3 -1 5 5 6 - 7 5 4 9 1 + 2T+\7-5 - 6 - 8 1 + 5 3 16 6 -0"-016 - 5 - 25 2 + 2 2 29 - 4 + 33 3 9 1 68 2V -I 3 + 10 [ 1 > 11] 4- 91 -13 -12 - 0"-001 - , 2 _ 2 4- 1 4- 5 3 27-2D - 0"-002 2 + 64 -11 2 + 1 1 4 3 - 127 -10 3 2 11 4 7 - 9 6 2V-1-21 T -18-2 3 76 5 1 - 8 8 - 5 - 0" 004 4 + 3 - 7 13 -6 - 83 1 + r-88 6 22 -7 + 3 27- 20 + ^-88 - 0"-002 - 5 39 4 - 0"-004 1 8 - 4 87 2V + 1 2 + 13 - 3 716 - 9 + 0" 001 27-21 ) + 2T+ 17- 3 + 6 2 + 152 -8 + 2 6 + 0"-065 4 + 8 1 + 74 -7 + 2 5 4 + 39 -6 + 2 37-2D 6 2 1 13 -5 4- 3 3 -0"-003 2 4- 10 -4 4- 1 7-^+88 3 + 7 -3 4- 4 27 - 4D 5 - 0"-001 4 + 5 -2 + 4 3 + 0"-008 4 - 2 5 + 3 -1 + 3 3 5 6 + 2 + 2 -4D 2 13 1 + 1 3 + 0"-007 1 + 8 2D-, I + T-50 4- 2 -1 + 0"-002 2F-2D 3 - 0"-002 3 -0"-007 2F-2D + r-90 5 + 0"-054 2F - 2D + 1 + T - 90 5 + 0"-003 ADDENDUM. FINAL VALUES 91 Final coefficients of sines of i (M - T) + arguments at top of each column. i 2D- -J/ + 149 2 ^) = Z I +0' '019 -5 + 0"-003 -6 -0"-001 2 + 8 -5 3 3 + 3 2D- 2M+ 117 -4 17 4 + 2 -6 - 0"-002 3 + 5 5 + 1 -2 + 2 1 Jf+212-7 -1 - 0"-002 2D-Z -Jf + 149 -1 +0' '002 -7 + 0"-001 + 5 1 + Jf +212 -6 + 3 1 + 60 1 + 0"-007 -5 + 38 2 - 10 2 1 -4 3 3 - 4 5 + 8 -3 1 4 2 5 - 1 l- - M+ 147 2D- 1-2M+ 117 -2 + 0"-001 -6 - 0"-030 2M + 63 -1 7 1 -0"-002 21-2D 2 12 Z-t -2M +63 4 - 0"-002 3 + 5 2 - 0"-001 4 + 2 6 + 6 21-'. 2D + M+2U" 5 + 1 5 + 0"-015 l- -2M-63 3J/ + 96 -2 + 0"-001 2^- 2D + 2Jf + 63 3 -0 "002 6 -0"-019 4 + 1 6 - 0"-001 92 ADDENDUM. FINAL VALUES Final coefficients of sines of i (J T) + arguments at top of each column. i I 2D-Z 21 1 + 0"-069 -3 + 0"-001 -4 -0"-004 2 -0"-013 2 13 2 + 4 -3 20 -1 + 8 -2 804 21 + J-T ft /"-6-2 -1 + 7 2 + 0"-002 -0"-209 1 8 + 3 1 + 11 2 171 2/-2D 2 + 8 2D-/ + J-100 2 -0"-187 + 15 -3 +0"-004 2,7-18 3 + 0"-002 2^-2D + J-( -0"-009 2D-/ + ,7-7 2 -0"-190 l-\ 7-6-7 - 2 + 0"-007 2D -0"-021 2l-2~D + 2J- - 5 + 0"-001 1 + 2 2D -1-J+ 6-73 2 -0"-002 4 + 2 2 + 58 -4 -0"-001 -3 + 3 3 1 -3 - 6 2/-4D 2 45 - 2 + 306 2 + 0"-009 -1 + 2 I - J+ 6 + 2 - 1 -0"-002 OT\ 7 7 Qr\ 1 ' ~^ C' ~~ t/ OU 2l-4I> + J- 1 + 21 - 1 + 0"-005 2 -0"-006 2 2 l- .7+ 106 2D-/- 2,7+18 31-2D 2D + .7-6 3 -0"-008 - 2 + 0"-009 2 -0"-007 -0"-002 4 2,7-18 2D-Z-2J+107 3^-2D + J-"t 2D -,7+7 2 + 0"-002 - 1 + 0"-002 2 -0"-005 2 + 0"-020 + 2 h + J- h" 2D + 1 2F-2V -0"-004 -2 -0"-003 2 - 0"-002 2 - 1 2F+1-2D 2D+1-J+7 2 -0"-002 - 2 + 0"-001 4D-^ 2 + 0"-001 -2 -0"-007 4D-^-^+7 -2 +0"-003 IRSITY OF ERRATA. Page 10, 13, 14, Page 10, 13, 14, Page 14, Page 15, Page 16, Page 19, Page 69, Page 75, line 7 For - -007066 i 2 - -008148 i 3 read + -006624 1 2 + -008260 i 3 . -001210/ 3 read + -008260 i.- -001238 i. line 8} 8 For --008148t 2 line 3 from foot, For - -02161 read - -00812. line 7, For - -4753 read + -4458. lines 4, 5, For round its centre of mass read round the centre of mass of the earth and moon. equation (18), For E= read R = re&l part of. line 2, For 3D -F + / + 2T-3Q read 3D-F- 1 + 2T- 3Q. line 11, For +722 read +726. -0"-007 -0"-003. -147 -148. Page 77, column headed

-2 read + 2F. Page 80, line 1, For 3& + 24T read 3h + 22T. 4 from foot, 2nd column, For +0"-029 read +0"-027. 3 Delete $ + 1 +0"-002. ,, 5 For -0"-0152 read -0"-0086. last line, For -0"-013 read + 110 +0"-007. CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ASTRdNGNY jWs^ "N^ LD 21-100m-ll,'49(B7146sl6)476 S