MR. LUBBOCK ON THE THEORY OF THE MOON. 125 



portant result may also be arrived at very simply by means of the method of deve- 

 loping R, I gave formerly*. 

 Employing the same notation, 



dy dyrdy~'~dA'dy'dsdy 



i2 = m,|-^--^{l +3cos(2X^-.2y -2*2} 



- g^ {3(1 — 4 *2) cos (X' — X,) + 5 cos (3 X' — 3 \)}. 



It is evident, by mere inspection of the value of R, that the term in question, of 

 which the argument is 3 r — i-j- 3 1^ — 2 ;j, can only arise from the development of 



-g^cos(3X^-3X,), 

 5 r * 



so that we may consider /? = — ^-^ cos (3 X^ — 3 Xj in the present investigation, and 



dR ^ 



ds 



The argument -J- 3r — |4-3|, — 2 ri, can only be made up, therefore, of the argu- 

 ments 3 r — I -f 3 ^^ and 2 rj, and 3 r -|- 3 1^ and | + 2 ??, by subtraction, if we limit 

 ourselves to that part of the coefficient which is multiplied by e ef y^. 



It is therefore necessary to determine the coefficients of R corresponding to the ar- 

 guments 3 r + 3 1^ and 3 r — | -|- 3 1^. 



By the expression 



dR _ dR dy; d fi d \ 



de^ ^ d r^ ry d ^y ' d A^ d ^y' 



it is evident that the coefficient of cos 3 r -|- 3 I, depends only upon the coefficients of 

 cos 3 r, cos 3 r -f- 1^, and cos3 r -f 2 l^. On reference to the development of RX, it 

 will be found, that considering only these terms, 



iJ = ^'|-4cos3r + |-eyCOs(3r+y-^ey2cos(3r-l-2yj, 



[116] [120] [128] 



and since 



dR dR ,dR dR 



^, TT" = «/ T~> and XT' = 'AZi 



r being used for nt — n^t. 



;rd7^= - cos|,--2e,cos2|,- -g-e;cos3i„ 



j^' = 2 sin ^, + ^ e^ sin 2^, + -^e^ sin 3 1^, 

 neglecting terms which are not required. 



* Philosophical Transactions, 1832, p. 606. 



t I use the letters r, ^, g^, and ij, where formerly I used t, x, z, and y. 



X Philosophical Transactions, 1831, p. 266. 



