LAPLACE. 555 



Accordingly we find that Laplace does in effect return to the 

 subject ; see his pages 363 — 366. 



In the formula which we have given in Art. 697, suppose 

 a= 0, and 5 = 1; then if the event has been observed to happen 

 m times and to fail n times out oi m + n trials, the probability that 

 the chance at a single trial lies between a and yS is 



J a 



I x'^il-xydx 

 Jo 



-f m T \/(2m7i) ^ m T \/(2mn) 

 Let a = , p = — 1 — , 



where fjL = m + n; then we shall shew, by using Laplace's method 

 of approximation, that the probability is nearly 



2 



[V«V^ (2). 



VttJo 



For with the notation of Art. 957 we have ?/ = £c"'(l -xY; 

 the value of x which makes ^ a maximum is found from the 

 equation 



m n 



= 0, 

 SO that a = 



m + n 

 Then 



Y 



e = log 



^ {a-\-ey{i-a-ey 



~2 K+(l-a)^J ~3 I? ■"(13^1 + - 

 Thus, approximately, 



2 _ 1/ ) "t It, ] 6" 



2 ja" ' (l-O 2»m 



