156 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



identical with that of enumerating the lines of route which neither cross nor touch 

 the line AL, for each such line of route gives a permutation tf the letters in a m y8" 

 which is required by the conditions. 



I prefer in the first instance to alter the conditions of the problem so as to 

 determine the probability that Pierre never at any instant has fewer votes than 

 Paul. The lines of route to be enumerated are then those which do not cross, but 

 which may touch the line AL. 



Owing to the different interpretations that may be given to the line of route many 

 courses of procedure are open. I select that one which in the special graphs gives the 



partition 



752221 



and, in general, a partition having TO for the highest part and a number of parts not 

 exceeding n+l. 



Regarding zero as au admissible part, let the parts of such a partition be (in 

 descending order) 



,, 2 , ... n + 1 . 



These parts are subject to the conditions 



a, > a 3 2 , 



i> 3 + 2 (,,), 



.f S: 4 ( 4 ), 



!2:a4 + 3 (a s ), 



<- ! 2: - (t(w-t), 



*1 2: Ot n ,-\-n' 1 (2n'-3J, 



where 11! = n+l. 



We can perform the summation 



^'Za*... #,"' 



for all numbers satisfying the above 2// 3 conditions. 

 9. Suppose n = 3, the sum in question is 



a, 



the meaning of the symbol | being that, after expansion in ascending powers of 



Xj, x 2 , x 3 , all terms involving negative powers of a lt a 3) a 3 are to be rejected, and that 

 in the surviving terms a lt a 2 , a 3 are, each of them, to be put equal to unity. 



* See "Memoir on the Theory of the Partitions of Numbers, Part II,," 'Phil, Trans.,' A, vol. 192, 

 1899. 



