LAPLACE. 539 



The sum of the coefficients of every power of t up to infinity 

 in the expansion of u will represent the probability that the play 

 will terminate if there be no limit assigned to the number of games. 

 But the sum of these coefficients will be equal to the value of i6 

 when t is made equal to unity ; and this value of u is unity. Hence 

 we infer that the probability of the termination of the play may 

 be made as near to unity as we please by allowing a sufficient 

 number of games. 



983. In Laplace's own solution no notice is taken of the fact 

 that equation (1) does not hold for x = n. Professor De Morgan 

 remarks in a note to Art. 52 of the Theory of Probabilities in the 

 EnciJclopcBdia Metropolitana, 



Laplace (p. 240) has omitted all allusion to this circumstance ; and 

 the omission is highly characteristic of his method of writing. ISTo one 

 was more sure of giving the result of an analytical process correctly, and 

 no one ever took so little care to point out the various small considera- 

 tions on which correctness depends. His Theorie cles Probabilites is by 

 very much the most difficult mathematical work we have ever met 

 with, and prmcipally from this circumstance : the Mecanique Celeste has 

 its full share of the same sort of difficulty; but the analysis is less intri- 

 cate. 



984. We may observe that as Laplace continues his discussion 

 of Waldegrave's problem he arrives at the following equation in 

 Finite Differences, 



1 



i/r, X yr-\ , x-1 "f" ~e)n U i\ x-n ^ j 



in integrating this, although his final result is correct, his process is 

 unsatisfactory, because it depends upon an error we have already 

 indicated. See Art. 955. 



r 



985. Laplace's next problem is that relating to a run of 

 events which was discussed by De Moivre and Condorcet ; see 

 Arts. 325, G77 : this problem occupies Laplace's pages 247 — 253. 



Let 2^ denote the chance of the happening of the event in a 

 single trial ; let </> {x) denote the probability that in x trials the 



