596 LAPLACE. 



We integrate next with respect to z^ ; thus we shall remove z^ 

 entirely, and introduce the factor 



that IS, ^- 



LLiJlT 

 ^ 1 1 h 



where -^ = —2 + 





2 



3 



and the first four terms in u^ are replaced by the single term 



\^a 2 2 

 ^2 , ' 2^7 2 J that is, by /x'2;/. 



A + «4 t>3 



By proceeding in this way it is obvious that we shall arrive at 

 the assigned result. 



1029. Laplace devotes his pages 391 — 394^ to a problem 

 which we have indicated in Art. 911. The problem resembles 

 that which we have noticed in Art. 1027, and the mode of solution 

 will be illustrated hereafter in Art. 1036. 



The problems which Laplace considers in his pages 385 — 394 

 relate to the probabilities of future events ; and thus these pages 

 are strangely out of their proper place : they should h.2iYQ followed 

 the discussion which we are about to analyse in our next Article, 

 and which begins thus, Considerons maintenant la prohahilite des 

 ivenemens futurs, tir^e des evenemens observes. 



1030. Laplace considers in his pages 394 — 396 the impor- 

 tant subject of the probability of future events deduced from 

 observed events : see Ai'ts. 870, 903, 909. 



Retaining the notation of Art. 1021, suppose that z, which is 

 a known function of x, represents the chance of some compound 

 future event depending on the simple event of which x represents 

 the chance : then the whole probability, P, of this future event 

 will be given by 



I yzdx 



_ 



j y dx 



J 



Laplace then suggests approximations for the integrals in the 



