316 Prof. Darwin, On the perturbation of a comet [Mar. 



But r cos eo = p cos + R, hence the disturbing function is 



irjI_I_4 C0S 



[p R R> 

 Differentiating with respect to p and 0, we have 



P = -M \\ + —cos e 

 [p M 



T=^sm0. 

 Hence F 2 = M 2 jl + -1 + JL cos 1 



But C' = 5+JL\ 



r 



^72 J^-2 ^4 (■ 2 4 "J 



and thus _ = ^_ y!? |l + 2 ^co S « + t.|. 



Again the disturbing function for the motion of the comet 

 round Jupiter as perturbed by the Sun is 



S \- +-775COS 

 [r R" 



It will be seen that the sign of the second term is here + , 

 because the angle between R and p is ir — 0. 



In this formula we have 



r 2 = p 2 +R 2 + 2pR cos 0. 

 Plence differentiating with respect to p and 0, we have 



*® = S.Rfp-^\ sin 0. 



Now we might proceed to square these two and add them 

 together to find jf 2 , and so go on to find the rigorous expression 

 for Jp/CD, which equated to F'/G will give the rigorous equation 

 to the required surface ; but the result would- be so complex as 

 to be of little value because not easily intelligible. 



I therefore at once proceed to approximation. 



