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.

22 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO
The complete generating function can be obtained without difficulty, but, on account
of its great length, I restrict my endeavours to the establishing of the 13 ground
products. I find it necessary to adopt abridged notations, and in future, where it is convenient, I denote
NalV,2a2i3a31AT4a4N5a, by (alC2a3a4a5).
Further, if a portion of the generating function presents itself, which involves rnerely
ground products already obtained in the previous work, I enclose it in brackets [ ] and
thenceforward omit it. For example, I write
A = [B] + C C = [D] + E E;
and so on.
^ (n +1\^ fn +- ^2 fn<~a + 34 In a 5\^
For (+ 1) (-t 2) (- S ) ( 4) ( )
\-2 3 4:5
= laN2aiVyN44Na s = (ala ).
When n is of form condition is
4p + 1, al + 2a3 + a5 > a2 + 2a4,
4p + 2, 2> a4,,
4p + 3, 2al + -a3 + 2a > a2 + 2x4,
3p + 1, a2 + a, > as,,3p + 2, a1 + a4 > a3,
5p + 1, a4 > as,
5pr +2, a3 > a5,
5p + 3, a2 > ac omit,
5p + 4, aa > a.;
the eighth of these conditions may be omitted as being implied by the second and
sixth. I separate the generating function into six portions corresponding to
Case 1. a, > a, a2, 3;
Case 2. ai a2, a3 > a,;
Case 3. a2> a, Ca2> a, aI+a5 a3;
Case 4. a2 >al, a2,>3 3 >, + a5;
Case 5. a,>a, a3 >a2, alc+a,5 a;
Case 6. a2 > a, cas>a2, a3 > a> + -5.