THEORY OF THE PARTITION OF NUMBERS. 
671 
where 
\ > X and. X —}— yu, —(— 1 > X! -j~ /a • 
These are enumerated by 
Hence 
x q Jc 2 ( x ). 
^2 ( X ) — /1 — 
Therefore 
(1 — x) (1 — x~) 1 — x 
1 
+ + x'fc (*). 
^2 (*) — 
(1 - x) (1 - x*y (1 - a?) 
and the number of specification graphs containing two graph lines is 
x 6 & 3 ( x ) or 
(1 — x) (l — & 2 ) 2 (1 — X s ) 
Similarly we shall find that k 3 ( x ) is composed of four parts corresponding to the 
occurrence of 0, 1, 2, or 3 specification graph lines. The first three are readily seen 
to be enumerated by generating functions 
x~\ (x) 
_ x G k 2 (x) 
(1 _ ®) (1 - a: 2 ) (1 - a?) ’ (1 -x)(l- x s ) ’ 1 -x ’ 
When three specification graph lines occur, the form must be 
2^+3 
2^'+2 j «' 
2A" + 1 p" 
\ \ and X + /a + 2> \ + /x + I > a -(-p, 
and the generating function x l2 Jc 3 ( x ). 
Hence 
h i x ) = 
(1 — (1 — # 2 )(1 — a? 3 ) (1 — x) (1 — £ 2 ) 
i ' ^2 C y ) i io 7 / \ 
+ 77- ^ ^ + T3W + x *3 fo), 
and we can show that 
^3 ( x ) = 
(1 - a) (1 - ® 2 ) 2 (1 - a?) 2 (1 - a 4 ) 
