=17
given on fage 7, and let any one of the conditions on page 13 de¢
satisfied, and let all the s's converge, Then II[14+ v¢(x)].1+):c'¢'
where the C's are given by formula (11).
. As a special case, let vy(x) = ayx . If Joy = s, is abso~
lutely convergent, then all the s's converde absalutely, by Theorem
I; if the a's are real numbers and if sy and s, converge, thean all
the s's converge, and except for s,, the convergence is absolute."
In either of these cases, Fv (x) converges for all values of s,
and equals syx, For every finite R, the convergence is uniform for
]x] <R. Now all the conditions of Theorem IIla are satisfied if
lagx| < 1-€, The C,'s become ¢,'s and formula (11) takes the more
simple form*®
g (.1)2(1(-1)7‘4

! R
1 s il g

(11%) Co =ty g Sis Je ¥

;A g
Myleee "q! Jad leve qu

where the summation is taken fer the values of the indices which

satisfy the equation ZIm¢ji = w .

Fundamental Theorem of Symmetric Function Theory,

The hypothesis of Theorer IIIb permits the choice of one
of the conditions on page 13. It was shown on pages 11-12 that if
lvg) <1-¢, then %u;hx" 'converges in Ry, so that if we chcose
condition 2, we have added to the hypothesis of Theorem IIIb oniy
that 3|v¢(x)| converge uniforuly. As mentioned on pages 13-14, this
condition implies.that IT[1 + »4(x)] is absclutely-uniformly conver-
gent in R,. Therefore the value of II is not changed by any permu=

tation of the rows of the array (1) and is a symmetric function of

e e et eeeeetsees ettt etm S et e e
17 Brouwich, seo. 18, Smeil, seo. 126.

1% ) derivation of formula (11') is given for a finite number of elements, with

a proof by mesns of symmetrio funotions, in Serret, Vol, 1. sec. 3198-203. The fore

muls is given with suggestions for deriving it, in Macmehon, Vol.I, Chap,I, and in

Macmahon, Intro, Chap.I, but only for a finite number of elements, Formuls (11) is

given in Macmshon, Vol.II, p. 282-3 for the special case of two sgpquences,