ii2 The Rev. Thos. P. Kirkman on 



k parts is x, any number >o of the parts beginning the 

 partition being zero (as e.g. a x — a 2 = a 3 = o, the rest being >o) ; 

 next, having named clearly our k dark diagonals as I st , 2 nd , 

 3 rd , . . . /£ th , we have to insert into the I st split diagonal, the 

 first a x ( >o) of the x primes in the belt; next to insert into 

 the 2 nd split diagonal the second number a 2 >o of the 

 remaining x—a x primes in the belt, and so on, lifting and 

 dropping the primes in their order till all the k dark diago- 

 nals have been, by the guidance of the same partition, 

 a 1} a 2 , . . . a k , charged each with a given number a £ >o, of 

 primes, leaving none in the belt. 



We have to make the like use of the next k-partition of 

 the number x to empty by its guidance the same full belt, 

 by charges >o placed in every split diagonal, till every 

 k-partition has been so used to empty the same full belt. 



If the number of the k-partitions of x so used is m, we 

 shall have turned the same irreducible H into m different 

 partitioned R-gons. We shall presently see what number 

 vi is, and be able to describe them. But our task is only 

 begun by this handling of all the m k-partitions of our 

 number x of primes in our belt, each in dictionary order. 

 We have to handle in the same way, for the charging of 

 our k dark diagonals of H, every permutation of the parts 

 of each of those m k-partitions, emptying our full belt into 

 the split diagonals M times ; where M is the number of 

 all the permutations of the k-partitions of x\ which M 

 includes the number m of the k-partitions above handled 

 unpermuted. 



It is plain, that each of these M permutations bids us 

 distribute in a different way our x primes into our /'-split 

 and unsplit dark diagonals, of which the z th will be unsplit» 

 when the z th place of the guiding permutation is zero. But 

 all this fuss of distribution is mere wind. M is all we really 

 want, and that theorem Q. gives at once. 



5. It is then evident that the number of distributions of 



