b

Each line of this sum can be written a¢ gqu , where the accent
indicates that b¢ is omitted from the sum. Let R = §|b!, LS
$01 is absolutely convergent, then %:'lb,,l SB, and |pq,] § I;::a&:'b,]
Egldtl E:llbll £ M. From this we find that $11 is absclutely
converéent. By breaking #4k...q into its component sums and by
repeating the process just used, we can show that #spe..q -is
absolutely convergent for every choice of j, k, ***, g, provided

$1, po1, toos, °°°°, are absolutely convergent.

We have now proved the following theorem:

TrEOREM I, Given the sequemces (ag), (bg), (cq), *°°v, (1g),**,
A mecessary and sufficient condition that all the s's and all the p's
shall comverge absolutely, or shall be symmetric, is that s,, So1,
so01, *°**, or ta, foi, Poos, *°°°, Comverge absoluteily, or are
symsmetric, dbsolute convergence and symmetry are equivalent coxdi~
tioms of the ¢'s and the s's; either one may be replaced by the

other,

Expansion of an infinite product in terms of the p's,
and in terms of the s's, and other fundamental formulas.

In the next paragraphs some fundamental formulas will be
derived by means of which we can express symmetric fupctions in
terms of the #'s or the s's, together with the conditions under
which these expressions represent the functions,

Let x be a complex variable, and let &, B, v, «==« A, «-,
be a set of distinct positive integers arranged in ascending order
of magnitude. Now form the series °
vi(x) = a;xa + b;;fl f: c(,xY 4 veeseees 4 111)‘ 4oveeee i=1, 2 3, <wee- .
With this set of v's we shall form the infinite product
I = {1 + v4(x)] . Various convergence conditiocns will be im~-

posed on the v's and II as we proceed. The exponents o, B, y,---,