1 30 REV. T. P. KIRKMAN ON THE 



and that if Xi be confined to any value f, which it can 

 receive in (A), the number of solutions of (B) is that of 

 the {k — l)-partitions, as they are called, of N — a — Jif, 

 Now Xi may have any value from zero up to the greatest 

 integer in (N — a) : k. Hence it appears that the A-partitions 

 of N, none of which contains a part less than a, are equal in 

 number to the {k — l)-partitions of N — a+the (k — l)-parti- 

 tions of N — k — a + the k — 1-partitions of N — 2k — o, &c., as 

 far as k can be continually subtracted to leave a remainder 

 not negative"; no part, in any of these (k — l)-partitions, 

 being less than a. Take a=l, and let j^Vx denote the 

 number of A-partitions of x\ then the theorem just proved is 

 expressed thus, if 



X=:ke'\'Cy x—k=k(€ — \)+c, cvO, z.k-\-\, 



fcPx=A-lPa:— l4"fc— lP^-l-A;+A;— iP^-l— 2&4"* • • • +A;-lPc— 1> 



whence follows 



(C) 



fcP^r— ftPa; k=k-l P^r-l (D) 



From this equation of differences ^P^ can be found. It is 

 easily seen that 



,P,=K«-*2.-.), (E) 



Where *2^_i is that well known function of the square roots 

 of unity which is = 1 for x odd, and = for a? even. To 

 the reader who is not familiar with these circulators, as they 

 are called, it is sufficient for our purpose here to define, that 

 ±a**5e in this paper is to be read as ±a, when the quotient 

 e\s is a positive integer, and as ± 0, when e:s is either frac- 

 tional or negative; where it is to be remembered that 0:*, e 

 being zero, is always counted whole and positive. The last 

 equation affirms that the bi-partitions of x are \x in number 

 when X is even, and ^(x—X), when x is odd, the truth of 

 which needs no demonstration. By equations C and D, if 

 x=Se-\'C, c being yO, = 1, 2, or 3, 



