LAPLACE. 509 



Now [ ^] ^^[1 ■"-) J hence in u( h) the coefficient 



of fr^ will be h't'' (-jiA > provided we suppose that ?/ is made zero 

 after the operation denoted by 8'' has been performed on -^ . 



Similarly in w (^ - aj the coefficient of fr' wiU be oTA' (^/] , 

 provided we suppose that x is made zero after the operation de- 

 noted by A'" has been performed on -^ . 



In this way we obtain 



+ ^ V A (^] + — ^ F A'^ (^] 



IT K f ( X A 



+ • • • + 7 : 7T- r y^y. A I — ^ 



Thus we see that in order to obtain z^^y we must know 

 ^0, 1 > ^0, 2' • • • ^P ^^ ^0, V > ^nd we must know z^^^, ^2, c • • ^p to z^^ „ . 



Now we have to observe that this process as given by Laplace 

 cannot be said to be demonstrative or even intelligible. His 

 method of connecting the two independent variables by the equation 

 generatrice without explanation is most strange. 



But the student who is acquainted with the modern methods 

 of the Calculus of Operations will be able to translate Laplace's 

 process into a more familiar language. 



Let E denote the change of x into x + 1, and F the change of 



7/ into ?/ + 1 : then the fundamental equation we have to integrate 



will be written 



{EF- aF- bE- c) z,^, = 0, 



or for abbreviation 



EF-aF-bE-c = 0, 



Then E'^F^ will be expanded in the way Laplace expands 

 and his result obtained from E^'F^z^^^. Thus we rely on the 



foundations on which the Calculus of Operations is based. 



