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.

MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS, ETC. 13
otherwise the theories proceed on widely divergent lines; as might be expected the
arithmetical products form a more extended group than the algebraical.
Denote, for brevity,
1 - -xns,n + s
and
1 - xS s
by Xs and Ns respectively.
The principal X theorem, that has been obtained loc. cit., is to the effect that constructing any X rectangle
X1 X2 X3 X4... X
X2 X3 X4 X5... Xl+
X3 X4 X... X+2
X4 X5 X6 X7... Xi+3
Xm Xm+l X+,2 Xm~+3... X +-1,
i and m having any values, with the law that any X has a suffix one greater than
the X above it or to the left of it, the product
X1X22X33... Xl+n-,
obtained by multiplying all the X's together, is finite and integral for all values of the
integer n. There are other forms as well, e.g. the product
XlX,XX3X4X,
which are not expressible in the rectangular lattice form, the theory of which is not yet
complete.
We see therefore that the product of N's contained in the rectangle
N1 N2 N,...- N
N, NV3 N,4... N1+1
V3 N4 N5... lY1+2
ATm N,,m~ Nrn~3 N... NI+n _
is an integer for all values of n.
It will appear moreover that no product exists which is free from NI, so that all
these products, being irreducible, are fundamental solutions of the problem.
The method of partition analysis is concerned with the solution of one or more relations of the type
XiCat + X.2 + X33 +... + Xsas
~ k11 + /13232 + 13/33 +... + t3t,
the coefficients X and,u being given positive integers, and it is required to find the
general values of Ca, a2.../3,,32,..., being positive integers, which satisfy the one or
more relations.