438 BICQUILLEY. 



814. We will briefly indicate the steps for the solution of the 

 problem in which we require the probability that n — S white balls 

 shall never occur in A. 



Let (j) {oc, n), </> {x, n — 1), (/) [x, n — 2) represent the number of 

 favourable cases in x trials, where the final number of white balls 

 in A is 01, n — 1, n— 2, respectively. 



Then we have the following equations 



(f> (x + 1, n) =(j) (x, n — V), 

 (f){x+l,n-l)= ?i'</) (x, n) + 2{n-l) ^ {x, ?j - 1) + 4(/) {x, n - 2), 

 <j>{x^l,n-2) = (n-rf<^{x, n- 1) + 4 {ii - 2) ^ i^x, 7i-2). 



If we denote </> (x, n — 2) by u^ we shall arrive by elimination at 

 the equation 



w^+3 ~ {Qn - 10) u^^^ + (Sti' - 16?i + 12) ii^^^ + W (?i - 2) u^ = 0. 



Then it will be seen that <f>(x, n — V) and (x, n) will be ex- 

 pressions of the same form as </> (x, n — 2). Thus the whole num- 

 ber of favourable cases will be aa""' + h^'' + 07""', where a, h, c are 

 arbitrary constants, and a, ft 7 are the roots of 



z' - {6n - 10) z'' + (Sn' - 16?i + 12) z + W (n-2) = 0. 



815. A work on our subject was published by Bicquilley, en- 

 titled Die Calcul des Prohabilites. Par C. F. de Bicquilley, Garde- 

 du-Corps du Roi. 1783. 



This work is of small octavo size, and contains a preface of 

 three pages, the Privilege du Roi, and a table of contents ; then 

 164 pages of text with a plate. 



According to the Catalogues of Booksellers there is a second 

 edition published in 1805 which I have not seen. 



816. The author's object is stated in the following sentence 

 from the Preface : 



La theorie des Prohabilites ebaucliee par des Geometres celebres m'a 

 paru susceptible d'etre approfondee, et de faire j)artie de renseignement 

 ^lementaire : j'ai pense qu'un traite ne seroit point indigne d'etre offert 

 au public, qui pourroit enriclier de nouvelles verites cette matiere inte- 

 ressante, et la mettre a la portee du plus grand nombre des lecteurs. 



