DANIEL BERNOULLI. 233 



Operations the black balls will be probably distributed thus, ii^ in 

 the second urn, v^ in the third, and ic^ in the first ; similarly the 

 red balls will be probably distributed thus, ii^ in the third urn, v^ in 

 the first, and w^ in the second. 



It should be observed that the equations in Finite Differences 

 and the solution will be the same whatever be the original distri- 

 bution of the balls, supposing that there were originally n in each 

 urn ; the only difference will be in the values to be assigned to the 

 arbitrary constants. Nor does the process require n white balls 

 originally. Thus in fact we solve the following problem : Suppose 

 a given number of urns, each containing n balls, m of the Avhole 

 number of balls are white and the rest not white ; the original 

 distribution of the white balls is given : required their probable 

 distribution after x operations. 



419. Daniel Bernoulli does not give the investigation which 

 we have given in Art. 417. He simply indicates the following 

 result, which he probably obtained by induction : 



3 



together with similar expressions for v^ and w^. These can be 

 obtained by expanding by the Binomial Theorem the expressions 

 we have given, using the known values of the sums of the powers 

 of a, P, y. 



420. Now a problem involving the Differential Calculus can 

 be framed, exactly similar to this problem of the urns. Suppose 

 three equal vessels, the first filled with a white fluid, the second 

 with a black fluid, and the third with a red fluid. Let there be 

 very small tubes of equal bore, which allow fluid to pass from the 

 first vessel into the second, from the second into the third, and from 

 the third into the first. Suppose that the fluids have the property 

 of mixing instantaneously and completely. Required at the end 

 of the time t the distribution of the fluids in the vessels. 



