132 MR. LUBBOCK ON THE THEORY OF THE MOON. 



When the elliptic values of the coordinates are substituted in the disturbing func- 

 tion, the term in question arises only from the expansion of the quantity 



3 7-' 



+ 4 r 3 *^ 



and in the elliptic motion 



/ 



2 Q 



f= - ^ cos 2 ?? + y2 ^ cos (I — 2 ;j) — -^ y2 ^2 ^os (2^—2;?) + &c. 



2 



^77 — ~4.8"~4"'"4.2.2.2~~ 16* 



Writing the index between brackets instead of the cosine of the corresponding 

 argument, in order to save space. 



-/^'■^ 



+ ^me3 — ^me^2^ [24] - j^m^ e^ e^[25] + -^me^e^[27] + jg m2 c^ e^ [28] 

 -^m2ee/[30] +^m2ee/[34] +^m2e;[36] + "^f [37] +6-4m2en39] 

 + ^m2e4[40] +^m2e3e^[42] +|| m2 e3 e, [43] - ^^m^ e^ e^ [45] 

 4- gg m^ e^ e^ [46] ^^ ^^ ^^ ^i^ [48] + j^m2e2e^2 [-52] __^2^^^3[54] 



1 /] 1 *! Q4')T 741 



-^gm2e.3[55] --3^m2e;[57] -3im2e;[58] +~^m^ej^[60] -{^^m2e/[70] 



I have verified some of the terms in the expression for the reciprocal of the radius 



vector given by M. Plana, which depend on ^, and arise from the second portion 



of R ; very few, however, of these can be obtained without a further development of 

 the disturbing function, in consequence particularly of the term 



j^ e^ cos (r + y , 



