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L. On the Variation of the Semi-axis Major of the Moon's Orbit, 

 By JOHN WILLIAM LUBBOCK, Esq., Treas. M.S. F.R.A.S. and 

 F.L.S., Vice- Chancellor of the University of London, fyc. 



DOISSON, in his Memoire sur le Mouvcment de la Lune autour de 

 -* la Terre, has considered the following theorem, that the expres- 

 sion for the variation of the semi-axis major contains no argument 

 of long period, accompanied by a multiple of m less than m*. In 

 this paper he has taken into account the terms which arise from 

 the second approximation, or that in which the squares and pro- 

 ducts of the quantities 8, 8 a, $e, 8co, 8y and may be neg- 

 lected. It is evident that terms may arise in the next, and in- 

 deed in every succeeding approximation of the order m 3 , which 

 must be taken into account in order to prove the proposition with 

 sufficient generality. Thus the variation of the eccentricity con- 

 tains terms multiplied by m, as of the argument 9, (2 T 2 ) ; 

 these, multiplied by others of the order m*, give in 8 e* terms of the 



order m 3 . and these multiplied by . . give in -y d t terms 



J d e* d c * dc 



multiplied by wz 5 , which after integration become of the order m 3 ; 

 if the argument being of the kind under consideration, the divisor 

 introduced by integration is of the order m*. It is true that Poisson 

 refers to his Memoire sur la Variation des Constants arbitrages 

 Mem. de I'Academie, torn, i., for an extension to the third approxi- 

 mation, that is, to terms depending upon the cube of the disturbing 

 force. In this paper, however, expressions are employed, which 

 take for granted that the disturbing force can be exhibited in the 

 same form, developed in terms of the initial values of the coordi- 

 nates x, y, #, and of the initial values of their first differential co- 

 efficients T--, V^ 5 T-. This development has never been exe- 

 dt dt dt 



cuted, nor has it been shown to be possible. When this system of 

 constants is employed, the quantities which are equivalent to [a, ], 

 [, e] t &c., become equal to unity and rigorously constant, so that 

 it is unnecessary to consider the effect produced by their variation. 

 Even, however, with the simplifications which recourse to this pe- 

 culiar system of constants affords, Poisson admits that this demon- 

 stration would become too complicated to admit of its extension to 

 the higher powers of the disturbing force. M. de Pontecoulant has 

 made objections to the proof given by Poisson, but as he differen- 

 tiates* in a manner at variance with that intended in the expressions 

 of which Poisson makes use, and does not allude to the Memoir 

 in question, in which a further approximation is attempted, these 

 objections are not to be considered as exactly identical with those 

 of that distinguished astronomer. 



* Conn, des Temps, 1840, p. 21. 



