410 SIR j. F. w. h'erschel on the algebraic expression of the 
identical in number with those of .r— 7, beginning with (1, j:’— 8), and so on. Thus 
we have 
®n(a?)=^n(.r— i)+^n(a’— 4)4'^n(x — 7)+ &c. 
Next, with respect to the number of terms to which the right-hand member of this 
equation is to be continued : it will be that of the columns, which will continue without 
repetition so long as the number x—^lm in the first combination (m, m,x — 2m) of 
any one of them shall be not less than m, or so long as x—?>m shall not be negative. 
CC OC 
Hence we must have m— =, since the next greater value of m, viz. m==-l- 1, will give 
3 3 
^ 00 00 00 00 
the tripartition ( = + 15 = + 1? ■3^—2 = — 2). Now x cannot exceed 3.= by more than 
\ o O O / o 
00 00 00 
2, so that x—2 = ~2 cannot exceed =, and must therefore be less than = 4-1. Hence 
3 3 3 
we conclude that the number of tripartitions is derived from that of bipartitions by 
the equation 
l)-l-^n(j: — 4)-{- 7) — to= terms. 
o 
(27.) Applying a similar reasoning to the higher cases, we shall find as follows : — 
n(.'r) = n(^— l)-|-n(j: — .s— l)-f-n(a? — 2^ — l )....= terms; .... (42.) 
a relation which, with many others of greater generality, has also been arrived at by 
Mr. Warburton. 
(28.) Suppose now we set out from the equation *n(j?) = 1, and proceed to derive from 
this value those of ^n(^), ®n(a?), &c. in succession. It will be apparent from the course 
of the foregoing investigations, and from the nature of circulating functions, that the 
general expression for n(x) must consist of two portions, the one non -periodical, a 
function of x, and which may be represented by (p{x), the other periodical or circu- 
lating, which we may denote by Q^, so that we shall have in general to consider the 
s-1 
following form of n(a?), 
n(,r) = <p(x)-l-Q,, 
s 
from which to derive the value of n(,r). 
When we substitute this in the general expression (equation 42.), we get 
n(x) = 9(j’— ])-}-^(a?— 5— l)-]- (3/ terms) 
+Qa-i + Qx-s-i+ (3/ terms), 
00 
where 3/= =. Now with respect to the first portion of this, if (p{x) in any one case be 
a rational integral function of x, it will be so in all subsequent cases, as is evident 
from the course of the preceding investigations. This part of n(a?) then has been 
already dealt with, and its complete expression is X-f-Y of equations (34. 35.), or 
(38. 39.). 
