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.

18 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO
and the compound syzygies
(Nr) (C) - (1NT2N3) (B) = O,
(NN22NV3) (B) - (NNV231 ) (A) = O;
indicated by the numerator terms:
-N12N22N3, - N2N1N3V, - IV12N24z2, + N3Y2V4NA32, + N13V25V32
respectively.
The generating function takes also the suggestive form:1 - N ON:l3NV
1 - N. 1 - N1N,. 1 - NWN2N3. 1 - NXN23N
+ 1 - NN. 1 -N113.T2N23
1-nN. 1-N1N2N3. 1 -Nrl3N231N3
By proceeding in this manner we not only obtain the new ground products appertaining
to the order but also those of lower orders previously obtained. It would be desirable to
exclude the latter, and in the case before us we see cà posteriori that this could have been
secured by impressing the additional condition
al = 33;
but no method, similar to this, seems to be available for an order higher than 3, as no
equation invariably connects the indices of the ground products.
ORDER 4.
+)l (n 2a2 (f + a23) (n + 2aa I+4)~ N 4 N V N3aN
When n is of the form condition is
4m + 1, a, + 2a3 > 4- 2a4,
4m + 2, a2 >a,
4m + 3, 2ax + a3 > a2 + 2a4,
3m + 1, 2 ass
3m + 2, ai + a4 > ~3
The Q function which can be at once written down is somewhat troublesome to deal
with, so that I find it appropriate to divide the generating function into two parts according
as ai >C3 a3 > al.
Case 1. ai > a3.
The conditions reduce to
cal a3 (a),
cl + 2a3 > a2 + 2a4 (b),
C2 > a4 (c),
a > a3 (d),