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.

16 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS T
shewing that the ground products are N1, N1N2,
or (ala2)=(1, 0); (1, 1).
ORDER 3.
(1z + la) (nn + 2)2 (f + '' al
When n is of form condition is
4m + 1, a, + 2ac3 > C (a),
4m, + 3, 2a1+ a3, c
3m + 1, a2 a3 (b),
3m + 2, a > ac2 (c).
We may omit the second of these as being implied by the first and fourth and introducing
the auxiliaries a, b, c in the relations marked (a), (b), (c) respectively we write down the
fI function
Q 1
b a,.
1 - acN. I -b. - -
a bo
as the expression of the sum NallNl2aN3%.
It must be observed that the operating symbol Qf, has reference to each of the three
auxiliaries a, b, c.
These must be dealt with in the most convenient order, so that unnecessary labour
may be avoided; this order is not always obvious without some preliminary experiments. In
the present instance it is clearly advisable to commence with b because a occurs to the
second power, and operation upon c will introduce a3. It should be remarked that operation
upon one letter nmay cause two letters to vanish; this would indicate that the relations
associated with these letters are not independent members of the system of relations.
It does not follow conversely that if the relations are not all independent two letters
must vanish as the result of operation upon some one letter. This does follow for a
certain order of operation upon the letters, but not for all orders.
Eliminating b we obtain
2 1
1 a
1 -acN. 1 - N2. 1 -_ 2N
Ca C
Observe that this expression would have presented itself if for the two relations
a1 + a3 a2
ao > a3,
we had constructed the sum N1ciala22 (j\T<N3)a.