510 LAPLACE. 



We may notice tliat we have changed Laplace's notation in 

 order to avoid the dashes which are difficult in printing. La- 

 place uses X where we use y, t! where we use r, and 'A where we 

 use 8. 



953. Laplace takes another equation in Finite Differences. 

 The equation we will denote thus 



A"^,„+^A»-a^,.,+ J A-^ax, + ... = 0. 



Here A belongs to x of which the difference is unity; and S 

 belongs to y of which the difference is a. 



Laplace says that the equation generatrice is 



e-r*K'-r(i-)4.&-r&-r--=»- 



He supposes that this equation is solved, and thus decomposed 

 into the following n equations : 



t a\ W 



t a V TV ' 



where q, q^, q^}"* ^^^ the n roots of the equation 

 Then, using the first root 



u 



Then passing from the generating functions to the coefficients, 

 that is equating the coefficients of ^V°, we obtain 



