DANIEL BERNOULLI. 231 



416. The next memoir by Daniel Bernoulli is entitled Dis- 

 quisitiones Analyticce de novo prohlemate conjectwrali. 



This memoir is contained in the Novi Comm...Petrop...Yo\. 14, 

 1769, pars prior. The date 1759 occurs by mistake in the title- 

 page. The date of publication of the volume is 1770. The 

 memoir occupies pages 1 — 25 of the part devoted to memoirs. 



417. The object of the memoir is to illustrate the use of the 

 Differential Calculus, and it is thus analogous to memoirs which we 

 have already noticed by Daniel Bernoulli. 



Suppose three urns ; in the first are n white balls, in the second 

 n black balls, in the third n red balls. A ball is taken at random 

 from each urn ; the ball taken from the first urn is put into the 

 second, the ball taken from the second is put into the third, and 

 the ball taken from the third is put into the first ; this operation 

 is repeated for any assigned number of times : required the proba- 

 ble distribution of the balls at the end of these operations. 



Suppose that after x operations the probable numbers of white 

 balls in the three urns are denoted by u^., v^y w^ respectively. Then 



1t/~.,t — U,f ~~ ~1 • 



"'■^^ "^ n n 

 For — is the probability of drawing one white ball out of the 



n 



10 



first urn, and -^ is the probability that a white ball will be drawn 



n 



from the third urn and so put into the first. Similarly 



By eliminating, using the condition u^-\-v^-\-w^= n, we may 

 obtain an equation in Finite Differences of the second order for 



Ujc, namely, 



/^ 3\ /^ 3 3\ 1 



'^x^'i = "^^ar+l (2. ] —f^'x 1 H— )+-• 



x+2 x+i \^ ^J X \^ ^^ ^^y ^ 



But the following process is more symmetrical. Put w^^^ = Eu^, 

 and separate the symbols in the usual way ; 



