LAPLACE. 483 



Laplace says that the integral in the numerator is to be taken 

 from ic = to u = x, and from a? = to ic = 1, and that the integral 

 in the denominator is to be taken for all possible values of x and ii. 

 Thus putting u — x = s the denominator becomes 



[ [ u'' (1 - u)' s^' (1 - sY du ds. 



•J {^ ■J Q 



Laplace's statement of the limits for the numerator is wrong ; 

 we should integrate for x from to u, and then for u from to 1. 



There is also another mistake. Laplace has the equation 



V_ £_ ,JP_ l__ = n 



X l-X'^X-x l-X-^x 



He finds correctly that when a? = this gives 



X = 



jy+p 



He says that when jr = 1 it gives X = l, which is wrong. 



Laplace however really uses the right limits of integration in 

 his work. His solution is very obscure ; it is put in a much clearer 

 form in a subsequent memoir which we shall presently notice ; see 

 Art. 909. He uses the following values, 



p = 251527, q = 24.1945, 

 ;/= 737629, ^' = 698958, 



and he obtains in the present memoir 



410458 ' 

 he obtains in the subsequent memoir 



1 



P = 



410178* 



The problem is also solved in the Theorie . . . des Proh. pages 

 381 — 384 ; the method there is different and free from the mis- 

 takes which occur in the memoir. Laplace there uses values of p 

 and q derived from longer observations, namely 



. ^ = 393386, ^=377555; 



31—2 



