160 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



12. Hence the number of partitions having a highest part m and n+l parts, zero 

 being included as a part, subject to the given conditions as regards magnitude is 



fn + ml\ _ fn + ml 

 ( n J ( n-2 



which may be also written 



m n+ 1 /m+n\ 



m+l \ m )' 



This, therefore, is the number of lines of route which do not cross the line AL. 

 Hence the probability that Pierre is never in a minority is 



mn+l 

 m + I 



13. From this probability, which call F (m, n), is immediately derivable the 

 probability disciissed by BERTRAND and ANDRE, which call P (m, n). 



For in the lattice - - is the probability that the line ol route passes through the 



point M, and thence we find 



p / \ m T7 { \ m ~ n 



m + H m + n' 



Other probability questions may be discussed in a similar manner, with the 

 advantage that light is at the same time thrown upon several other problems of 

 partitions, compositions, and combinations of unipartite and bipartite numbers. In 

 the above investigation we have had before us partitions of unipartite numbers which 

 have a given number of parts, a given highest part and parts which in addition satisfy 

 certain inequalities. 



14. If we had had before us the parallel theory of the compositions of unipartite 

 numbers there would have been the composition 



in correspondence with the partition 



the weight X/3 and the number of parts n would have been given and the parts 

 ^8,, ft?,... would have been subject to the inequalities 



A 2= 2, 



A 



