556 LAPLACE. 



Therefore 



/ 



x'^il -xydx Y e-^'df 



jx'^il-xydx yJ e-t'dt 

 = -r- I e'^'dt= -J-- I V^ dt. 



"We have thus two results, namely (1) and (2) : the former is 

 obtained by what we may call an assumed inversion of James 

 Bernoulli's theorem, and the latter we may say depends on Bayes's 

 theorem. It will be seen that the two results are not quite con- 

 sistent ; the difference is not practically very important, but it is 

 of interest theoretically. 



The result (2) is in effect given by Laplace on his page 366 ; 

 he does not however make any remark on the difference between 

 this result and that which we find on his page 282. 



On page 209 of his RecJier cites... sur la Proh. Poisson gives the 

 result (1) which he obtains by the same assumption as Laplace. But 

 on his page 213 Poisson gives a different result, for he finds in effect 

 that the probability that the chance at a single trial lies between 



m V \/{2mn) , w [v -{■ dv) \J{2'mn) 

 — — - and '. 



is Vdv, 



where F=-r- e"^' ^7;:^ ^-e"^' (3). 



This is inconsistent with Poisson's page 209 ; for if we take the 

 integral j Vdv between the limits — t and + t for v it reduces 



to — / e~^ dt, so that we arrive at the result (2), and not at the 



result (1). It is curious that Poisson makes no remark on the dif- 

 ference between his pages 209 and 213 ; perhaps he regarded his 

 page 209 as supplying a first approximation, and his page 213 as a 

 more correct investigation. 



Poisson's result (3) is deduced by him in his RecheTches...sxir la 

 Proh. from the same kind of assumption as that by which he and 



