THEORY OF THE PARTITIONS OF NUMBERS. 175 



If n candidates at an election have 



1, 2, 3, <* 



voters in their favour respectively, where 



and if any instant A 1} A 2 , ... A B _ l5 A B voters have recorded their votes in favour of 

 the several candidates respectively, the probability that ahvays 



s 



s=n tn 



n n (i ^ 



(=1 



The interesting point is the connection of the problem with the theory of the 

 two-dimensional partitions of any set of 2 unequal numbers. 



The graph solution proves at once the remarkable theorem that if (bJ>J) s ...) be the 

 partition conjugate to (a^cis...), 



a fact which it would be difficult to establish by pure algebra. In this case the 

 corresponding probabilities are in the inverse ratio of 



2 +a s +...)l 





