508 LAPLACE. 



Multiply the two series together. Let 



Fj = yld' + a; (c + al) a"'^, 



J- ■ ^ 1 . ^ 



Tr_ y(y-i)C y-2)„ . 



Then 



^^=4^f--^y+F.f--^rv...-^^. 



.T / VT 



T \T J \T J 



But the equation 



gives 



1 «j 5 



7 --c = 



CT T C 



1 



1 , c + a6 ' 

 



T 



therefore 



Now we pass from the generating functions to the coefficients, 

 and we pick out the coefficients of ^V° on both sides. This gives 

 z^y on the left-hand side, and on the right-hand side a series 

 which we shall now proceed to express. 



Let A apply to x, and indicate a Finite Difference produced 

 by the change of x into ic + 1 ; and let 8 similarly apply to ?/, and 

 indicate a Finite Difference produced by the change of 7/ into 

 2/+1. 



