158 THEORY OF JUPITER'S SATELLITES. [7 



or writing 



*-r-V 



A nJrl ~ A n = oA n , 

 /(a) = A 

 In the same way if we write 



we obtain 



/(a) = ^ jS 2 + SAft + /3 J [SM.a* + SvLa 4 + . . .] 



This series may prove more advantageous to deal with than the 

 original. In the case of the quantities I),, Legendre has given a trans- 

 formation which facilitates some calculations. 



Denote the series of coefficients 



s + i (s + i)(s + i+l) (s+j)_( s + i+i)(s + i + 2) 

 i + 1 ' (i+l )~(V +2) (7+ 1 ) (t + 2)"(i +3) 



by the symbols 



1, 1+A,, 1+2A. + A,, 1+3A.+ 3A. + A,, ... 



respectively, so that in fact A,, A,, etc., are the same as 8A lt ^A lt etc., 

 A l being unity. Substitute in the expression for l> t . Then within the 

 square brackets, we have the following terms : 



independent of A 



8 ,*(+!) 1 



1+ i a+ -TT- 2 a+ - = (T^*r 



multiplied by Aj 



^a- + g ( g+1 >2a- + ^ g+1 )^ + 2 ) 3 a" + JL -??- - 



1 1.2 1.2.3 (l-o 1 )*! -a" 



