Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

14 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO
This is accomplished by constructing the sum
xlx2 x3a,... Xsas yW y2y3... yt.t
for all sets of values of a2, a2.../3, /32,... which satisfy the relation. The expression
obtained is found to indicate the ground solutions of the relations and the syzygies that
connect them.
The sum is expressible in the crude form
Q 1
> 1 - rn'zx. 1 -rx2.m.... 1 - lw^ n Xs. 1 - m-ly y. 1 - y2... 1 - -nt~ yt
where the symbol of operation
Q
is connected with the auxiliary symbol m in the following manner:The fraction is to be expanded in ascending powers of x1, x2,... y, y2,...; all
terms containing negative powers of m are to be then deleted; subsequently, in the
remaining terms, m is to be put equal to unity.
Slight reflection will shew that the conditional relation will be satisfied in all
products which survive this operation, and that if we can perform the operation so as
to retain the fractional form we shall arrive at a reduced generating function which
will establish the ground solutions and the syzygies which connect them.
As a simple example of reduction which is of great service in what follows take
al > 1,;
Q 1
this leads to,
1- rx1. 1 -
and observing that
1
1-,~ - 1 x, 1 - x~y 1 m
Qi2M 1 1
we find =;
>1 -x,. 1 - x y 1 - l
m
Q mS X1S
also
1 — mx-. 1 — - yi
which is the solution of aL, /, + s;