November 14, 1907] 



NA TURE 



INEOUALITIES IN THE MOTIOX OF THE 

 MOON."- 

 'T'HE most interesting result of Prof. Newcomb's re- 

 searches on the planetary inequalities in the moon's 

 motion is that he has found i"-i4 as the theoretical 

 coelTicient of the Jupiter evection term. This term was 

 discovered empirically by Prof. Newcomb in 1876, and Mr. 

 Nfsill assigned its origin to Jupiter. Dr. Hill and Radau 

 independently computed its coefficient as o'-g. Two or 

 three years ago it was pointed out that the observations 

 indicated a coefficient i"-i, and now Prof. Newcomb has 

 obtained the same result by theory. This term is now 

 worked out ; the subject begins and ends with Prof. New- 

 comb, and has lasted thirty-one years. No indication is 

 given in the memoir before us as to why Hill and Radau 

 concurred in an imperfect value. It was a curious 

 incident, as we know of no other case where any result of 

 either of these two mathematicians has required revision. 



In order to illustrate the methods of the memoir, we 

 briefly indicate the process of calculating the Jupiter 

 evection term. 



First Stage. — Using the ordinary notation supplemented 

 by a = Iog-a, Prof. Newcomb starts from Delaunay, vol. ii., 

 pp. 235^ and 800, and obtains on p. ig the equations of 

 variation 



•^ at ?P, , ?P, , ?PA 



d a I a?.,^ a?, J ap,\ 



<i ; _ a[ ?Pi^ 8?!^ aPA 



,i at ?P, ^ ePi^ ?P,\ 



rf«/) ix\ da oe (ly J 



'i a «/??!_,_ 0Pi_L 9PA 



P, being the potential of the disturbing forces. 



He also gives (p. 18) the numerical values of a,, &-c., as 

 follows : — • 



0]= -F2-022S c, = 0'0l68 7,= -O'0229 



a.,= -0'0;,0[ e..= - I9'I534 7^= -O'020o 

 03= -HO 0075 ^3= + o'0026 7.;= - 5 '5700 

 Second Stage. — Neglecting certain small terms, we have 

 for the potential of the direct action of a planet 



" P] = MK U- - V-) JMC (^2 _ 3f2) + MD2I,, 

 and a similar form for tlie potential of the indirect action 



" Pi = - «-G(4- - 7,") - w-J(p2 - 3f2; + ,«2l2t7;. 

 M 

 For brevity, the sum of the two may be written 



" P, = io-^K'(|- - 7)-') - iQ-^Ci^- - sC") + io--iD'2|-j. 



M 

 In the above formulae, | ri, 5,', denote the lunar 

 coordinates divided by the moon's mean distance, the axis 

 of I being directed towards the mean sun ; 



p- = i- + -n" + P; 

 and the coefficients MK, MC, MD, m-G, m'J, nrl are 

 known functions of the positions of the earth and the 

 disturbing planet in their orbits. Of these six coefficients, 

 the expansion of the first three in the form 

 -COS ., '\ . ■ . 



I, j being integers, is exceedingly troublesome, and the 

 expansion of the other three assumes that the mutual 

 perturbations of the earth and planet have been calculated. 

 We are going to illustrate the methods of the memoir 

 by considering as an example the perturbations arising 

 from terms in Pj with argument 



2w — 2j or 2D — 2^-1-2^' — 2J. 



1 "Investigation of Inequalities in the Motion of the Moon produced 

 by the Action of the Planets." By Simon Newcomb, assisted by Frank E. 

 Ross. Pp. viii+160 (Washington : The Carnegie Institution, 1907). 



Such a term may arise in P, by combining a lunar argu- 

 ment 2D — 2^-1- j.g'' with a planetary argument 



-ig' + 2g'-2], 

 j being given any integral value ; but we will confine out 

 attention to the case ;' = o, which gives rise to the only 

 sensible term of the whole number. 



We require, therefore, to pick out the planetary terms 

 with argument 2J— 2^' = Nj, and the lunar terms with 

 argument 2D — 2^ = N. 



The following extracts from the memoir cover the first 

 part of the work : — 

 From p. 85, Table XXII. : — 



io''MK= -h6"'ii9cos N4 -1-0" -006 sin N4 

 Jio'MC= -o"-293cos N4-(-o"-ooi sinNj - 

 io'MD= -o"'oo5 cosN4 4-6"'ii4sinN4 



From p. 97, Table XXXIII. :^ 



io^?«2G= -24-668 cos Nj- o".35SsinN^ . 

 io^/«-J = - 8 '096 cos Nj- o"'o6isinN4 

 lo'w-I = + 0'363 cos N4 - 23" -488 sin N4 

 Hence by addition, p. 145, Table XLV., 



K'= -f3o"-8i cos N4 + (o"-38 sin N4) 

 C'=- 8" -39 cos Nj - (o"-o6sinN4) 

 D' = (-)-o"-35cos N4) -i7"-37sinN4 

 It will soon appear that the six terms of K', C, D' 

 fall into two groups of three ; one group of three is in- 

 dicated by brackets, and will not be proceeded with, as 

 the other and more important group suffices for illus- 

 tration. 



Third Stage. — With the notation (see p. 24) 

 f- - 7)'- = 2/ cos N 

 p-- 3r'=Z'7 cos N 

 2^7)= /-sin N 

 we extract from Table XL., p. 112, for this argument 

 yi = —2g + 2\ — 2\' = 2D — 2g=2ir — 2g', the values 



)''■'*— -1- n-rnn 30 2>= -HO'284 09 



2/= -1-0 '007 809 



2^= +0001 S07 

 /-= +0-007 185 



2i^= + O'COO 



2:."= +0-003 73 2V = +0-065 69 



ck 



= - O-QOO 60 



= +0-261 £9 



are 



and we note that the differentials w-ith regard to 

 insensible. 



These expansions are derived partly from Delaunay 's 

 lunar theory and partly from Brown's. 



Fourth Stage. — In the differential equations of variatior* 



put - P, = 2*cosN. 



0= - 2/ sin N.o„ where O(, = io, + i'a2+ /'oj 



(1 1', i", are the coefficients of I, v, 6 in N, or 2n-2g' , so 

 thn' i' = 2, i = i" = o). 

 Similarly, 



_' — e= - 2/ sin N. <'„ where c,, = it'i + iV., + I'V^. 

 dy/i/) 

 We shall drop the equation for 7, and extract from- 

 Table XLVn., p. 146, the values 



ao= - o-o6o2 ,-„= -38-307, 

 as may be easily verified from the values of 20, and 2e^ 

 at the beginning of this article. 



From Table XLV 1 1 1., p. 147, we extract values of pro- 

 ducts oi p q i k by a^ and e,, : — 



a„#= -0-000 235 0(,^= -O'OOO 0154 ia|,/-= -0-000 216 

 £„/>= -0-149 20 f„r7= -0-034 28 W'=-°''3790 

 ;\gain, noting that 



' 4;t -=- 2?/") + ,^2?? + 7, 2|^|cos N = 2L' cos N 

 \, da/ Ce cyi 



we shall drop the equation for 9„, and noting that when 

 q and h respectively replace p, then L', P' become in Prof. 

 Newcomb's notation L", P" and L4, P.j. 



d(iit) 

 a 



dUu)' 





NO. T985, VOL. ']l'\ 



