THEORY OF THE PARTITION OF NUMBERS. 
633 
and this is the coefficient of cU/3? in the development of the fraction 
_1_ 
1—« — /3 + (1 — fi) «/3 ’ 
(see “ Memoir on a Certain Class of Generating Functions in the Theory of 
Numbers,” ‘Phil. Trans.,’ Roy. Soc. of London, vol. 185 (1894), A, pp. 111-160), 
and this fraction may be written 
Hence 
a 9 /3 s 
(l - *y+i(i-/3) 
s+1 
a s /3 s 
(1 — «)* +1 (1 — j3) s+1 
is the generating function for the lines of route in all bipartite reticulations which 
possess s left-bends or s right-bends, and also for the other entities in correspondence 
therewith. In particular it enumerates all unipartite partitions into .s different parts 
limited, in any desired manner, in regard to number and magnitude. 
Art. 21. Various theorems in algebra are derivable from the foregoing theorems. 
The generating function for the partitions of all unipartite numbers into parts 
limited in magnitude to p and in number to q is 
_1__ 
1 — a . 1 — x . 1 — ax . 1 — ax 2 . 1 — ax 3 .... 1 — ax p ’ 
the enumeration being given by the coefficients of in the ascending expansion. 
The G.F. is redundant as we are only concerned with that portion, of the expanded 
form, which proceeds by powers of ax p . 
The foregoing theory enables us to isolate this portion, inasmuch as we know it to 
have the expression 
1 + 
n 1 
ax p -f- 
T 
9 
a~x 2p + 
J \ cdx'v + . . . . + ^ ^) cftotfP + 
which may be written 
(1 — ax p )- p ~\ 
As a verification of the simplest cases we find the identities 
1 
1 — a . 1 — x . 1 — ax 
1 
(1 — ax) 2 
2 1 ffi 
a 
+ 
x 
1 — a. 1 — x . 1 — ax . 1 — ax 2 
1 — a 1 — x 
1 l a 
ax t ax (1 — ax 2 ) 
(1 - ax 2 f [ * + 1 -a r 1 - x 1- a + l-a.l-ax 
a simple inspection of which demonstrates the validity of the theorem in these cases 
MDCCOXCVI.—A. 4 M 
