PREVOST AND LHUILIER. 455 



Now PnJ^n differs from P^ only in having p + r instead of p ; 

 and QnS,^ difiers from Q^ only in having q-\-s instead of q^. There- 

 fore the sum of all the terms of the form P„ QnP^n^n is 



\p + r\q + s I ??i + 1 



p-{-q + r-\-s-\-l \ni — p — q — r — s 



in — p — q 

 And .Y= ' ^ 



r -\- s m — p — q — r — s 



Thus finally the required probability is 



\r -\- s ' p + r \ q -{■ s \p -\- q + 1 



\IL\1 [rVL \l)-\-q-\-r + s-\-l ' 



844. Let us supf)Ose that r and 5 vary while their sum r + 5 

 remains constant ; then we can a^^ply the preceding general 

 result to ?' + 5 + l different cases; namely the case in which all 

 the r + s drawings are to give white balls, or all but one, or all but 

 two, and so on, down to the case in which none are white. The 

 sum of these probabilities ought to he unitif, which is a test of the 

 accuracy of the result. This verification is given in the original 

 memoir, by the aid of a theorem which is proved by induction. 

 No new theorem however is required, for we have only to apply 

 again the formula by which we found S in the preceding Ai'ticle. 

 The variable 2')art of the result of the preceding Article is 



}:)-{- r \q + s 



that is the product of the following two expressions, 



(r + 1) (r 4- 2) p) factors, 



(5 + 1) (5 + 2) q factors. 



The sum of such products then is to be found supposing r + 5 

 constant ; and this is 



p-\-q + l r + 



Hence the required result, unity, is obtained by multiplying 

 this expression by the constant part of the result in the preceding- 

 Article. 



