7] THEORY OF JUPITER'S SATELLITES. 



157 



II. 



DEVELOPMENT OF THE DISTURBING FUNCTION. 



In the mutual disturbances of the Satellites hereafter considered, the 

 disturbing forces are obtained from a function of which the expression 



[a 2 - 2aa' cos (nt -rit + e- e') + a' 2 ]"* 

 is the chief part. Write 



a/a' = a, nt n't + e e' = 0, 

 and let us consider the development of the expression 



according to cosines of multiples of <. We may evidently take a to be 

 less than unity. 



Write 



S " = (1 2a cos 4> + a 2 ) * = - b a + ^ cos < + &., cos 2<f> + . . . , 



A 



then our investigation deals with the coefficients 6. 

 Now 



develops S~" by the Binomial Theorem and pick out the coefficient of 

 00si^ = -(e & * + e~**) ) and we get 



Zi 



, _(*+!).. .(s + t-1) 4 r. ss + i s(s+l)(s + i)(s + i+l) 1 



bi = 2 -- ~-r -a 1 1+-^ a 2 + -' - v , . '\ r . ' a 4 + ... . 

 1 . 2...i [_ li+l 1.2 (fc-fl)(t+2) J 



Such a series as this may be transformed with advantage in certain 

 cases, so as to proceed by powers of a different quantity to a. For 

 example if 



f(a)=A 1 

 then 



