The K-partitions of the R-gon. 115 



the unplaced. But it is otherwise with our belt units. I 

 call them units because each has to contribute a unit to the 

 occupation of a place. To prevent confusion in our final 

 account, it is necessary, and fortunately it is sufficient, to 

 insist that if, directed by the first term of your guiding 

 permutation (any one of the above M), you begin by putting 

 (V) of my units in the place (the split diagonal) that you 

 have first chosen, you shall take the first cm. the belt before 

 you; and that if you take d more to put into your second 

 place, you shall take the next d in the belt ; and, moreover, 

 that my units in their new place shall sit in order, and wear 

 their marginal triangles, exactly as they did in the belt. 



The diagonal split has a name, 12, 13, or ac ; where 

 a < c . This a is in the lowest contour of the split ; which 

 must match the lower contour of the belt used. 



7. It is here also important that we fix and name 

 exactly our five places. 



Call them Fig. H, afiydz ; 12 = a, 13= j3, 34 = 7, H — $, 

 56 = g; and let the lowest point of each, standing vertical 

 before you, be the above 5 first figures, 1, 1, 3, 1, 5. 



We are to split 12 = a from 1 up to 2, and to separate 

 the halves enough to admit between the parallels 12 and 

 i'2' the portion of JSi that we are inserting, which inserted 

 shall read from left to right, as it reads in the belt 3Bi. 



And so exactly for 13 = j3, &c. 



This splitting and distributing is all imaginary work ; 

 whatever be k and x, the sum M, which is all we really want, 

 of the k-partitions of x and their permutations, is given by 

 my theorem £1. Vide page 2 1 1 of the Proceedings of this 

 Society, 1892-3, where in line 9, ab infra, k Q x is to be read 

 for *Q. 



The k-partitions of the number x here meant are those in 

 which both zero parts and repeated parts are permitted, but 

 not negative parts. 



