484 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
multinomial coefficients {cf. Cayley, loc.cit.). This is done because it is the universal 
practice in the theory of the single system, and because otherwise it appears to 
possess undoubted advantages. 
The symmetric functions which appear in the relation are fundamental since, as will 
appear, they serve to express a,ll other rational integral symmetric functions, and they 
may be further termed single-unitary, in that, not only is each composed entirely of 
units, but also each bipart comprises but a single unit. 
It is obvious that the number of biweights connected with the weight w is 
tv + 1. 
5. It may be asked in how many ways it is possible to partition a biweight into 
biparts. 
In the ordinary theory of partitions the number of partitions of a number iv is the 
coefficient of in the ascending expansion of 
_1_ 
1—x.l — I — x^.l — 
In the present case, the number of partitions of the biweight pq into biparts is the 
coefficient of' xPy'^ in the ascending expansion of 
_1_ 
1 — X . \ — y .1 — x^ .1 — xy .1 — y'^ .1 — x^ .1 — x~y. 1 — xy"^ . \ — y'^ .. . 
or, putting ij equal to x, we see that the whole number of partitions of the weight 
V -k q into biparts is the coefficient oi xp^‘^ in the ascending expansion of 
1 
(1 (1 (1 _ 
Further, it is clear that the number of partitions of the biweight into exactly 
p biparts is the coefficient of a^xPy'^ in the expansion of 
1 
I — ax. \ — ay . \ — ax ?. 1 — axy . 1 — ay^. I — ax?. I — ax'-y. 1 — axy“ . 1 — ai^ . . . 
6. It is convenient now to have before us a iist of the symmetric functions up to 
weight 4 inclusive. 
The expanded generating function is 
1 X y 2.x‘'^ + ^xy + + 3a:® + 4a:'^ 2/ + ^ “k 3^® 
-j- 5x^ + Tx^y + + 7xy^ + 5y* + . . • , 
and we have 
