LAPLACE. 505 



advantage of the notation for denoting powers, which leads him 

 to speak of Descartes and Wallis. 



Laplace points out that Leibnitz made a remarkable use of the 

 notation of powers as applied to differentials ; this use we might 

 describe in modern terms as an example of the separation of the 

 symbols of operation and quantity. Lagrange followed up this 

 analogy of powers and differentials ; his memoir inserted in the 

 volume for 1772 of the memoirs of the Academy of Berlin is cha- 

 racterised by Laplace as one of the finest applications ever made of 

 the method of inductions. 



951. The first Chapter of the first part oi Livre I. is entitled 

 Des Fonctions generati^ices, ct une variable; it occupies pages 9 — 49. 



The method of generating functions has lost much of its value 

 since the cultivation of the Calculus of Operations by Professor 

 Boole and others ; partly on this account, and partly because the 

 method is sufficiently illustrated in works on the Theory of Finite 

 Differences, we shall not explain it here. 



Pasres 39 — 49 contain various formulae of what we now call the 

 Calculus of Operations ; these formulse cannot be said to be cle- 

 monstrated by Laplace ; he is content to rely mainly on analogy. 

 LagTange had led the way here ; see the preceding Article. 



One of the formulae may be reproduced ; see Laplace's page 41. 

 If we write Taylor's theorem symbolically we obtain 



Ay.= v/^^-Vj/., 



where A indicates the difference in y^ arising from a difference h in 

 X. Then 



Laplace transforms this into the following result, 



2 



/ hd_ 



The following is his method : 



/d_ Y nh£/ hd_ h dY 



W^"^ -l)y. = e ^.^'^•^ Ve2 ci^ _ ^'2 dx) y^^ 



