MR. LUBBOCK ON THE THEORY OF THE MOON. 129 



r ' a ' ^ ' dr 



Let l*„ be that part of the coefficient of the nth argument in the development of the 

 quantity 



which corresponds to the argument of which'w is the index, and let R^ be the coeffi- 

 cient corresponding to the argument of which n is the index in the development of R, 

 R'„ the corresponding coefficient in the development of that part of ^ d 72 which is 

 multiplied by m, and only arises in the second approximation, with its sign changed, 

 then the quantities r„ are given by equations similar to the following, 



ri I /l + 3 e2 (l + ^\\ {2 - 2 7w}2 - 1 j = (2 - 2 my Vy 



Passing over terms given by M. Plana and arising from the first approximation 

 with which I agree, I come to r22. 



r22|{2 — 2m + 3c}2— 1 1= (2 — 2m + 3c)2r22 



_^r 2 + 3C ^^\^2R 



I (2 - 2 m + 3 c) ^ J 22 



Hi — 2 ''lO ~" i(j ^1 ^10 — 2 ^^ ''l — ^ -"22 — "~ 32 



_ 2 J . 3 . 7 2 Q^ , g . Q . 25 ^ _ 2125 ^ 



^22 — 24 . 2 . 2 ^ ""24.16^ + 24.32 ^ ~" 384 ^ 



M. Plana has ^g^ rn^ 



r25{{2-m + 2c}2- ij = (2 ~ m + 2 0)^X2, - 2 \^^^±^^+ ij m^ R^ 



f> _A - ^ 2 » _-! 



% — 2 ''is ''la — ~ 32 ^ 25 — Q 



16.3.33 „ 2.2.3 ^ 7 „ ,, „ u 7 . 



'25 = - 15.2.32 "^ - TdZs ^' = - T ""'' ^' P^^^^ ^^^ "2- ''^ 



45 



{{2 - m-3cP - 1} = {2 - m - 3 C}2r45 - 2 {^4^4^ + ijm^ii,,, 



* Wherever I have found a disagreement with the result of M. Plana, as this might arise from an error in 

 my development of jR, I have verified the terms employed. 

 MDCCCXXXIV. S 



