672 
MAJOR P. A. MACMAHON ON THE 
and the specification graphs containing three graph lines are given by 
x 12 & 3 ( x ) or 
x- 
.12 
(1 - x)(l - X*f (1 - X 3 y (1 - X*) 
In general we obtain the relation 
1 
km (*®) — 
+ 
x*k y (x) 
(1 — x) (1 ~ x 2 ) ... (1 — x m ) (1 — x) (1 — xF) ... (1 — x m ~ l ) 
st?Te z (x) £ s(s+1 )Jc a (x) 
(1—x) (1—a; 2 ). .. (1— x m ~ 2 ) 
• • • + 
(1 — x) (1 — X 2 ) ... (1 — x m s ) 
— + .. . +x m(m+l) k M (x), 
and also 
7 , \ 1 *f" X" Jo -1 (x) -f" X^ (X -f* . . • "j” X?( sJrl) k s (.X s ) “f" ... 
k . =- ( i-.; )( i_, r s)(i-r 7 »)... -• 
where the numerator is the generating function for specification graphs of given 
content. 
Art. 63. We can now establish that k s ( x ) is the expression 
1 
(1 — x) (1 — & 2 ) 3 (1 - x?f ... (1 - xff (1 - x s+1 ) ’ 
for assume the law true for values of s equal and inferior to m — 1 ; substitute in 
the foregoing identity and writing 1 — x s = (s), 
(1) (2) . . . (m) (1 - x m2 + m ) k m (x) 
a; 3 (m) x & (to) (to —- 1) x mz- m ( m ) ( m _ 1)... (2) 
+ (1) (2) + (1) (2) 2 (3) + • ' • + (1) (2) 2 ... (m — l) 2 (to) ' 
Recalling the well-known identity 
1 
(1 — ctx) (1 — ax 2 ) ... (1 — ax" 1 ) 
(to) ax (m) (m — 1) 
= 1 + 7T7 • i — + 
a~x i 
+ 
(1) 1 — ax (1) (2) 1 — ax. 1 — ax~ 
( m ) (to — 1) (to — 2) 
a 3 .r 9 
(1) (2) (3) 
1 — ax . 1 — ax 2 . 1 — ax? 
+ • • . 
and putting therein a — x, we find 
1 
(2 ; (3) . . . (to + 1) 
X 2 (TO ) X 6 (ffl) (to - 1) 
(1) (2) (1) (2) 2 (3) 
X m "~ m ( wt ) ( m — 1) ... (2) ( X m ~ + m (TO) (TO — 1) ... (1) 
T" \ /o\o /*~>\o 77T7 i o\ /\ i” 
(1) (2) 2 (o) 2 . . . (TO - l 3 ) (TO) 1 (1) (2) 2 (o) 2 . . . (to) 2 (to + 1) 
