534 LAPLACE. 



978. After the solutions of these problems we have a few 

 pages headed Remarque sw les fonctions generatrices ; and this is 

 the part of the fourth Supplement with which we are chiefly 

 interested. It is here observed that in a case like that of our 

 Art. 975; the equation (2) is not an accurate deduction from equa- 

 tion (1) ; for additional terms ought to be added to each side, in 

 the manner of our Art. 973. 



There is however a mistake at the top of page 24 of the fourth 

 Supplement : instead of adding a function of t, two functions must 

 be added, one of t and the other of t'. 



The fourth Supplement then proceeds thus, on its page 24 : 



Faute d' avoir egard a ces fonctions, on j^eut tomber dans des 

 erreurs graves, en se servant de ce moyen pour integrer les equations 

 aux differences partielles. Par cette meme raison, la marche suivie dans 

 la solution des problemes des n"^^ 8 et 10 du second livre de la Theorie 

 analytique des Probabilites n'est nullement rigoureuse, et semble impliquer 

 contradiction, en ce qu'elle etablit une liaison entre les variables qui 

 sont et doivent etre toujours independantes. Sans entrer dans les 

 considerations particulieres qui ont pu la faire reussir ici, et qu'il est 

 aise de saisir, nous allons faire voir que la metliode d'integration ex- 

 posee au commencement de ce Supplement s'applique egalement a ces 

 questions, et les resout avec non moins de simplicite. 



The problem referred to as contained in No. 8 of the 

 TMorie...des Proh. is that which we have given in Art. 975; 

 the problem referred to as contained in No. 10 of the Theorie... des 

 Proh. is that which we shall notice in Art. 980. The fourth 

 Supplement gives solutions of these problems by the accurate use 

 of Generating Functions, in the manner of our Art. 973. 



Thus as Laplace himself attached the fourth Supplement to 

 his work, we may conclude that he admitted the solutions in 

 question to be unsound. We consider that they are unsound, and 

 in fact unintelligible, as they are presented by Laplace ; but on 

 the other hand, we believe that they may be readily translated 

 into the language of the Calculus of Operations, and thus become 

 clear and satisfactory. See Art. 952. 



979. We return from the fourth Supplement to the 

 Theorie... des Proh. itself. Laplace's next problem is that which 



