THEOEY OP THE PARTITIONS OF NUMBERS. 155 



6. The line of route also denotes the zig-zag graph of a composition of a unipartite 

 number. 



For placing nodes at all points passed over by the line of route we obtain 



. . r 







the graph of the composition 341122, and also of three other compositions 



221143 



12411211 



11211421 



of the number 13. (See ' Phil. Trans.,' Series A, vol. 207, pp. G5-134.) 



In general, we thus obtain four compositions of the unipartite number m + n+l. 



It will thus be noted that these four compositions of a unipartite number define two 



pairs of partitions of unipartite numbers, and clearly every theorem in partitions can 



be made to give a corresponding theorem of compositions. 



This manifold interpretation of the line of route through a lattice must be borne in 



mind throughout the following investigation. 



7. My object now is to show how certain questions of probability can be treated by 

 means of the lattice. 



BERTRAND and DESIRE ANDRE* have discussed a question which they have stated 

 in the following terms : 



" Pierre et Paul sont soumis a un scrutin de ballottage ; 1'urne contient m bulletins 

 favorables a Pierre, n favorables a Paul ; m est plus grand que it,, Pierre sera elu. 

 Quelle est la probabilite pour que, pendant le depouillement du scrutin, les bulletins 

 sortent dans un ordre tel que Pierre ne cesse pas un seul instant d' avoir 1'avantage ?" 



The probability is found by an ingenious method to be 



m n 



m+n ' 



8. I discuss the question by drawing in the lattice the line AL,t making an angle 

 of 45 with the line AB. The problem of BERTRAND and ANDRE is seen to be 



* 'Calcul des Probabilites,' par J. BERTRAND, Paris, 1888. J. BERTRAND et D. ANDRE, 'Comptes 

 Rendus de 1'Academie des Sciences,' tome cv., p. 369 et 436, Paris, 1887. 



t " Theories des Nombres," tome 1, par EDOUARD LUCAS, ' Le Scrqtin de ballottage' (pp. 83, 84, 164). 



x 2 



