-18..

the elements of (1). Theorem TIIb requires the convergence of =all
the s's. According to Theorem I all the s's converge and are sym-
petric functions of the elements of (1) if sy, So1, Soo1, °°°°, comverge
absolutely. All the ¢'s also converge and are symmetric functicns
of the slements of (1), so that if the conditions stated in this
and the preceding paragraph are satisfied, the hypothesis of
Theoremw I is alsc satisfied, Therefecre we have proved

ToeorEw IV, Let the elements of the array (1) be given
sweh that |vg(x)| < 1~¢, (C<e<1), nhen Q< x| <R, , and T|vg(x)| con~
verges uniformly in Ry, uhere vy(x) is as defined of page 7, and
such that sy, 301, Soo1, *°°°» converge absolutely. Then II[1 4 vy(x)]

converges absolutely~uniformly in Ry and equajls 1 + EC'z',uhcn

Ci=ty= 33,

C;=t21¢t01 = él'[(h! -~ 83) + (@s¢a)] ,
) Ca=ta+ t11+ foor = 51!'[(5:8 ~ 35152 + 5g) 4 ( 651501 = 6554) + (€500)],
(12
Ca = ta+ tzs + foz t ti0s + tocor = %’[(s,' —~ 633%s, + 8sysg—634+8s,%)

+ (12!;2501 - 245,511 + 24$91 " 1259’0,) + (12501: - 12801)
+ (As35001 = Asi01) + (Ase001) 1 «

In general the C's are given by forwula (3) and formuila (1D
and II, the ¢'s, the s's, and the C's are symvetric fumctions of
the elements of the array (1).

This is the fundamental theorem of the theory of symmetric
functions. In the remaining pertion of Fart I it will be supposed
that the conditions of this thecrer are satisfied,

When the weights &, 8, y, =+, are distinct positive inte-
gers as required in the definiticn of vy(x), all the choices of
these weights give essentially the same results. When the weights
are the set a=1,8=2, y=3, *»=*~, we shall say "the set 1, 2, 3,°**";
and for the set 1, 2, 3, “~~, we obtain the C's as given by fermulas

(12) atove. If now, for example, we wish to write the expressions

-