126 MR. LUBBOCK ON THE THEORY OF THE MOON. 



3 X niimei'ical coefficient of cos (3 r + 3 Q 

 — — 4X— gX— gXg— 4XgX — qXq 4X q^X IX^ 



+ 3X---g-X-4^X— Y+^X-g-X-g-X— -2- + 3X — 64X2X— "2 



- 340 + 240 — 20 4- 390 - 300 + 30 



128 



= 0. 



The first term of the coefficient, therefore, of cos (3 r + 3 |J in the development of 

 the disturbing function equals zero. 

 It is evident by the expression 



dR _ rdR dr dRdx 

 de drrde'dxde' 



that the coefficient of cos (3 r — i + 3 Q depends solely upon the coefficient of 

 cos (3 r + 3 1^) ; and as this equals zero, the other must also equal zero ; and as the 

 coefficient of cos (3 r — | + 3 |^ — 2 ?j) depends solely upon these two coefficients, it 

 must also equal zero, which was the point to be ascertained. 



M. PoissoN has shown in the memoir before referred to, that the first term in the 

 corresponding inequality of longitude depends only upon this coefficient in the deve- 

 lopment of R ; the inequality is therefore insensible. 



It follows equally that the coefficients of all arguments which result from any com- 

 bination of 3 r -{- 3 1^ with any multiples of | and 2 tj are also equal to zero. 



dr d a' 



Note. — The expressions for -t-t— and ^ I gave*, should be as follows : 



4^ = - I" + 1^ (1 - 4 O cos 2 ;j - y e cos (I - 2 ;?) H- y 6 cos (^ + 2 ;?) ^ 



i 



[62] [65] [66] 



13 

 ?) — -g y ^2 cos (2 1 + 2??; 



[77] [78] 



;; -|- y e sin (I — 2 ??) — y e s 



[62] [do'] 166^ 



IS 

 2 ;?) - -g- y e2 sin (2 l-f 2 ;?; 



[77] [78] 



3 13 



+ -g- y e2 cos (2 I — 2 ;?) — -g y ^^ cos (2 1 + 2 rj) 



2^ = — Y (1 ~ 4 e2) sin 2 ;; -I- y e sin (I — 2 ??) — y e sin (IH- 2 ;?) 



3 IS 



+ -g e2 sin (2 I — 2 ;?) — -g- y e2 sin (2 l-f 2 n). 



d R 

 The coefficient of the first term (argument 146) in the expression for -jr'}'? should be 



408 . ^ . - 204 



-j3^ mstead of -jg^-. 



* Philosophical Transactions, 1832, p. 606, f Ibid. p. 6. 



