218 
NOTE ON A PARTITION-SERIES. 
[826 
we find without difficulty (see infra) that 
l+P = (l+a®)(l + P), 
1 + P / + Q = (1 + ax 2 ) (1 + Q), 
1 + Q + P = (1 + ax?) (1 + P), 
1 + P + S — (1 + ax 4 ) (1 + S'), &c.; 
and hence, using il to denote the sum 
12 = 1+ P + ()(1 + ax) + R (1 + ax) (1 -1- ax 2 ) + >8(1 + ax) (1 + ax 2 ) (1 + ax 2 ) + ..., 
we obtain successively 
i2 -7- (1 + ax) = 1+P' + Q + P(1 + ax 2 ) +>8(1+ ax 2 ) (1 + ax 3 ) + ..., 
fi-f(l + ax) (1 + ax 2 ) = 1 + Q' + P + >8(1 + ax 3 ) + T(1 + aoc 2 ) (1 + ax?) + ..., 
i2 + (1 + ax) (1 + ax 2 ) (1 + ax 3 ) = 1 + P + S+T(1 + ax?) + ..., 
and so on. In these equations, on the right-hand sides, the lowest exponent of x is 
2, 3, 4, &c., respectively, so that in the limit the right-hand side becomes =1, or the 
final equation is 12 = (1 + ax) (1 + ax 2 ) (1 + ax?) ...; viz. we have the series represented 
by i2 equal to this infinite product, which is the theorem in question. 
One of the foregoing identities is 
1+P + S=(1 +cw 4 )(l + S'), 
viz. substituting for P, S, S' their values, this is 
1 i ax? t a 2 x? t a 3 x 15 t (1 + ax 8 ) a 4 x 2 - 
1 + X + Ï72 + ÎT2T3 + 1.2.3.4 
viz. this equation is 
ax? a?x 11 a s x 18 
= (1 + ax?) j 1 H—3—H ^—jr- + 
+ 
a* of 
1.2 1.2.3 1 1.2.3.4 ’ 
ax? — ax 5 (1 + ax?) a 2 x 9 — a 2 x n (1 + ax?) 
0^ + J + Ï72 
+ a s x 15 — a 3 x 18 ( 1 + ax?) ^ (1 + ax?) a i x 22 — a 4 # 26 (1 + ax?) 
that is, 
0 = — ax? + ax? — -j- + ^ 
1.2.3 
a 2 x 9 a 2 x? a 3 x 15 a 3 x 15 
1.2.3.4 
+ 
crÆ- 
+ 
a*x 2 
1.2 ‘1.2 1.2.3 1 1.2.3' 
In the same way each of the other identities is proved. 
Writing a= — 1, we have 12, =1.2.3.4...., 
= 1 +P + Q. 1 + P.l .2 +>8.1.2.3 + ..., 
where 
t> /1 \ (1+a? 2 )# 5 „ (l+x?)x 12 
P=-( l+x)x, Q = ±