SIMPSON. 203 



We have shewn in effect that the solution is obtained by taking 

 the coefficient of f~^^ in the expansion of 



a^ (1 — at) 

 (1-0 [I - t -^^ baH^'-'Y 



that is in the expansion of 



a^ (1 - at) 



{1-ty 





,T 1-at 1 (l-a)t 1 ht 



Now y, ^=- -+Vt— Tv.=^^ : + 



(i-ty i-t' (1-ty i-t ' (1-0'' 



We can thus express the result as the sum of two series, which 

 will be found to agree with the form given by Simpson, 



366. Simpson's Problem XXV. is on the Duration of Play. 

 Simpson says in his Preface respecting his Problems xxii. and xxv, 

 that they "are two of the most intricate and remarkable in the 

 Subject, and both solv'd by Methods entirely new." This seems 

 quite incorrect so far as relates to Problem xxv. Simpson gives 

 results without any demonstration ; his Case I. and Case ii. are 

 taken from De Moivre, his Case ill, is a particular example of his 

 general statement which follows, and this general statement coin- 

 cides with Montmort's solution ; see Montmort, page 268, Doctrine 

 of Chances, pages 193 and 211. 



367. We will give the enunciation of Simpson's Problem XX VI I, 

 together with a remark which he makes relating to it in his 

 Preface. 



In a Parallelopipedon, whose Sides are to one another in the Ratio 

 of a, 6, cj To find at how many Throws any one may undertake that 

 any given Plane, viz, ah, may arise. 



The 27th is a Problem that was proposed to the Public some time 

 ago in Latin, as a very difficult one, and has not (that I know of) 

 been answered before. 



We have seen the origin of this problem in Ai't. 87. Simpson 

 supposes that a sphere is described round the paralleleiDiped, and 

 that a radius of the sphere passes round the boundary of the given 

 plane; he considers that the chance of the given plane being 



li 



