232 DANIEL BERNOULLI. 



thus •i^-(^-D^'^"^''^' 



E-(l--] [ v^ = - u^, 



n I n 



{ 



^_(1__) [^^= ^;^, 



n n 



therefore \ E- ( 1 - - ) \ u^^iAux- 



nj I ^ \nj 



Therefore w, = ^ f 1 - - + -)\ ^ f 1 - - + -)V (7 fl - - + '^V, 



\ n nJ \ n nJ \ n nj 



where A, B, C are constants, and a, A 7 ^^^ the three cube roots 

 of unity. 



Then from the above equations we obtain 



therefore 



\ n nJ \ 71 7iJ \ n nJ 



Similarly 



\ n nJ \ n nJ \ n nJ 



The three constants A, B, C are not all arbitrary, for we 

 require that 



with this condition and the facts that 



Uo = n, ^0 = 0, Wq=0, 



we shall obtain A = B= G=-^. 



418. The above process will be seen to be applicable if the 

 number of urns be any whatever, instead of being limited to three. 



We need not investigate the distribution of the balls of the 

 other colours ; for it is evident from symmetry that at the end of x 



