172 THEORY OF JUPITER'S SATELLITES. [7 



in general, but 



1 7 a 

 b 1 -- T, 

 a' a'- 



where b i is that function of a or a/a' which we have been considering; 



and where a refers to the superior satellite and a' to the inferior, 

 taking a = a'/a which is less than unity, we have 



in general, but 



a 



Hence for the perturbations of an inferior satellite, we have 



1 db ; 



a , = -j-, 



da a da 



,dA- t 1 / dbi , 



-j- = --, a J- 



da a \ da 



we notice that A : is a homogeneous function of a and a', of 1 dimension. 



,, 1 , 



Also a 2 -W = -, a 2 ~ , 



da- a da 2 



, d 2 A, I I d 2 b t db\ 



aa' , -'-,= --, a 2 T- r l + 2a :r - I, 

 da da a \ da' da. 



a T-M -r 



' ' da 



In the case i=l, we must add to the differential coefficients of A it given 

 by the above formulae, the corresponding differential coefficients of a/a' 2 . 



For the perturbations of a superior satellite disturbed by an inferior 

 we have, where a = a'/a, 



a-r-= --T- 

 rfa a\ da 



a da ' 

 and so on. 



