604? LAPLACE. 



Laplace however adopts neither of the above methods ; but 

 forms a mixture of them. His process may be described thus : 

 Take the first form of solution, but use Bayes's theorem to deter- 



m, \^ 



mine the value of ^, instead of putting^ equal to i 



We will complete the second solution. The next step ought 

 to consist in evaluating strictly the integrals which occur in the 

 expression for P; we shall however be content with some rough 

 approximations which are about equivalent to those which Laplace 

 himself adopts. 



Assume, in accordance with Art. 993, that 



^ — (x'Y (1 - xy-- = - , 



where r is supposed to be not large, and to be such that nearly 



V = x^fjb — r, fjb — V = {1 — x^) [JL -^ r. 



Thus p= ^o^2^/;fa-^-'0 . 



I x'^^ii -xy^dx 



J 



Then, as in Arts. 957, 997, we put 



x^^ (1 - xY^ = Ye-^\ 



x = a+ ~-^^l , nearly, 

 where a = 



And finally we have approximately 



r2 



e ifia^il-a-i) 



^2'7rfid' (1 - a'O ■ 

 Then we have to effect a s-ummation for different values of r, 



