506 LAPLACE. 



Now let ^ ( -7- j denote any term arising from the development of 



I g2 dx Q~ 2dxj 



Then ,(^)....^^=,(^),,, 



and the term on the right hand may be supposed to have arisen 



/ hd_ _IlAY 

 Ve2 dx_ 



from the development of \e^ ^^^ — e ^ ^^J y^^'^- Thus the formula 



2 

 is considered to be established. 



We ought to observe that Laplace does not express the formula 

 quite in the way which we adopt. His mode of writing Taylor's 

 Theorem is 



and then he would write 



Ay,=\e "- -1). 



He gives verbal directions as to the way in which the symbols 

 are to be treated, which of course make his formulae really iden- 

 tical with those which we express somewhat differently. We may 

 notice that Laplace uses c for the base of the Napierian logarithms, 

 which we denote by e. 



If in the formula we put h = l and change x into a? — - we 



obtain 



/ 1 £, 1 d\' 



2 



which Laplace obtains on his page 45 by another process. 



952. The second Chapter of the first part of Livre L is entitled 

 Des fonctions generatrices a deux variables: it occupies pages 

 50—87. 



Laplace applies the theory of generating functions to solve 

 equations in Finite Differences with two independent variables. 

 He gives on his pages 63 — (j^> a strange process for integrating the 

 following equation in Finite Differences, 



