=R
where the surmaticn is taken for 4, j, +++¢, ranging fror cne tc
infinity, except that nc two elements are to be taken from the same
row of the set (1) and each pessible term occurs only once. For
exanple, the first term of pggy is aj,0,b3b,04cec¢,-

We can alsc forx the fellewing functions:

sy = J0g So1 = Tby BERY AR S33 = Jagby S1c1 = Jaq0y N e
Sy = Za‘* Sep = 2612 P $31 = 204251 Sseq = Fa‘ac‘ 5 Al vs
Sg = za‘fl Ses = fb(a I Sy = Za‘bia Sycq = Zfltcii Slh

s e s e * e e e s 0. e * s e o 0 e .0 e o 0 o .0 o o

In forring any of the terms cf the s's, all of the elements are to
be taken frew the sare rew of the set (1); that is, the subscripts
attached to the elements are all the sare in cne term. For example,
the first term of sgqs i8 a3%03%c;tf.

For bOti] the p's and.the s's the first subscript refers tc
the elements cf the first cclumr, the secend subscript refers tc
the elements of the secend cclumn, etc. FEach subscript denctes the
nusber of elements tc be taken fror the column and used as factors
c¢f a term. We shall speak of each cclumn cf the elements of the set
(1) as a sequence. Although each rcw can be called a sequence be-
cause it ccntains an infinity of elements which must be defined by
some law, the name will nct be applied to the rows.

The simplest symmetric functicns are these formed of the
elepents of one sequence cnly, with c¢nly a finite number of ele-
ments. The definiticn given in many algebras® is essentially as
fcllows: a symmetric function cf the elerents of a sequence ay, ag,
as, ****, en, is a function whose value is not changed by any per-
mutation whatscever cf the elements. In this paper we shall consid-
er functicns of infinitely wany sequences of elements, each sequence
containing an infinity of elements. Ve define a symmetric functicn

e ————————— et et ettt e e e

® DeComberousse, Niewenglowski, Weber. See slso Yacmahon. Rocher's definition is

more restricted, but he gives the eguivslent of our definition in a theoren,