-l

O

Taking 1 from each factor except two, we get all the terms

Sagayx™ + Sbibsx® 4 eqepa 4 eeene g $abss™*® 4 Sa5csa™Y 4 TogogBHY veene

% fflm + fio?’fl & fooaXiY 4 teeceseee 4 fiu"‘“e 4 hoi’aw * ¢°1‘xfi‘y vessve ;
Continuing this process we get each ope of the ¢'s exactly once,

pultiplied by the associated power of x, and we have®
Tom 14 pas™ 4 o™ 4 f0uf 4 £ox® 4 £330 4 poo ol 4 p s
% faxx’a’a 4 toazas + p10ax®™Y 4oveeens

Associated with #4a,..¢ is a power of x whose exponent is
J+ B+ °*c* 4 g\, that is, the weight of . Since we may have
2u=8, a+B=y, etc., the coefficients of some of the powers of x
way be grouped together, After this is done, the expapsion of the
prcduct may be written
) U= 14 Cox 4 Cox® 4 C,xa 4 ceveesey Cu,w"’ )
In this form, some of the U's may be zerc, for not all of the
powers of x need be present. The coefficient Cy of 5™ in the series
(2) is the sum of all the terms containing x¥ in the expansion of
the product II, No other terms cf weight w are to be found in the
series. Therefore, in terws of the ¢'s, Cy is the sum of all of the
#'s of weight w. FEach of the #'s is a symmetric function, and
therefore C, is a symmetric function, and it is of weight w. As a
general formula we can write
(3) Cow = Stipeeeq »
where the summation is taken for all values of the indices such
that jo + ¥ + **** + QA=w,

Since the weights of the ¢'s depend upon a, B, vy, sl
the representation of the C's in terws of the ¢'s also depends upon
«, B, v, **c= . The weight of fjheseq is 70 4 M 4 *ove ¢ o\, where
®, B, <==c, A\, j, k, ****, q, are all integers. In order to write

down the representation of the C's in terms of the ¢'s so that it

8

See Knopp, p. 487, sec. 258, Sats 1, for proof of this. The termg in the series
are arranged in the order in which they would sppear if @, 5, (0T, were 1, B