532 LAPLACE. 



provided we reject all terms in wliich the power of p^ surpasses 

 x^ — 1, in which the power oi p^ surpasses x^ — 1, and so on. 



Now on this process of Laplace's we remark : ' . 



First, the equation (2) is not true ; as in Art. 973 we ought to 

 allow for terms in which one or more of the variables x^,x^,x^,... 

 is zero ; and therefore additional terms ought to be placed in each 

 member of equation (2) of the present Article, like those in equa- 

 tion (2) of Article 973. 



Secondly, Laplace's treatment of his equation (3) is unintel- 

 ligible, as we have already remarked in a similar case ; see 

 Art. 952. By making use of the Calculus of Operations we might 

 however translate Laplace's process into another free from ob- 

 jection. 



976. At this stage we shall find it convenient to introduce an 

 account of the fourth Supplement to the Theorie.,.des Prohahilites. 

 This supplement contains 28 pages. Laplace begins with a few 

 remarks on Generating Functions; he gives the correct formula 

 for the solution of an equation in Finite Differences for which he 

 had formerly given an incorrect formula: see Art. 955. He does 

 not refer to the Theorie...des Froh. nor take any notice of the 

 discrepancy of the two formulae. He says, on page 4 of the Sup- 

 plement, 



Un des principaux avantages de cette maniere d'integrer les equa- 

 tions aux differences partielles, consiste en ce que I'analyse algebrique 

 fournissant divers moyens pour developper les fonctions, on peut choisir 

 celui qui convient le mieux a la question proposee. La solution des 

 ]>roblemes suivans, par le Comte de Lai^lace, mon fils, et les considera- 

 tions qu'il y a jointes, repandront un nouveau jour sur le calcul des 

 fonctions generatrices. 



We have therefore to ascribe all the rest of the fourth Sup- 

 plement to Laplace's son. 



977. The main part of the fourth Supplement consists of the 

 solution of problems which may be considered as generalisations of 

 the Problem of Points. There are three of these problems ; we 

 will enunciate them. 



