VII. Memoir on the Theory of the Partitions of Numbers. Part IV. On the 

 Probability that the Successful Candidate at an Election by Ballot may 

 never at any time have Fewer Votes than the One who is Unsuccessful; 

 on a Generalization of this Question ; and on its Connexion with oilier 

 Questions of Partition, Permutation, and Combination. 



By Major P. A. MACMAHON, F.R.S. 

 Received July 15, Read November 19, 1908. 





SECTION 1. 



1. CONSIDEE a lattice in two dimensions, taking, for instance, one in which AB, BC 

 are 7 and 5 segments in length respectively. It may be utilised for the study of 

 permutations, combinations, and partitions in various ways, also for the study of 

 certain questions in the theory of probabilities. 



L I C 



M 



2.* A " line of route " through the lattice from A to may be traced by moving- 

 over horizontal segments (a segments) in the direction AB, and over vertical segments 

 (ft segments) in the direction BC in any order. Thus one line of route is ADEFGHIC. 

 The number of such lines of route is 



or, in general, if AB, BC contain m, n segments respectively, 



\ 



m 



* See "Memoir on the Theory of the Compositions of Numbers," 'Phil. Trans.,' A, 1893. 

 VOL. CCIX. A 447. X 27.11.08 



