—o0

conditions imposed on the elements in Theorew IV, we obtain a
series of the form 1 + Cix + Cox® + Cgx® 4 *+°, where the C's are
expressed in terms of P's or S's by replacing the #'s in fornuli
(8) or the s's in formula (11). This is more clearly seen if we
write II[1+7'x 4+ 7,"x® 4 7" %® 4+ *+** 1, where the T's denote the
above quaptities in parentheses; and

Cam3N' =Py= Sy, ComXN'D') +300'= By 4 Fou = (5" = Sp) + (2,,)], ete
_From formyla (13) we obtain the expression of any P in terms of Sts,
which may then be expressed in terms of s's. The P's represent
syumetric functions of the most general 3 type, so that a linear
sum of these F's represents any symmetric I function of the most
general form. By use of the P's and S's we can therefore express
any symmetric I function in terms of the s's, Each of the s's can
be expressed in terms of the #'s, and therefore we® can in two geps
express any symgetric ¥ function in terms of the #'s. Without any
other proof we can state the ;asf theorem.

Turorew X. Every symmetric I fumction of any mumber of
iofinite sequences (ay), (bg), (cq), °***, satisfying the convergence
conditions of Theorem I and the conditions on the T*s above, can
be expressed im terws of the s's or the p's, the expressions being

found by means of the formulas as described above,

Examples:
(1) Say®™y =3(a®} (bl = Fy,(2,01) ,
By formula (18),  Pys = S;Sp; ~ 84y
By definition Sy = Z(a®)i =52, S,y = Zby = 2y, Sy = $(a)i (b = s25.
Therefore Zag sy = Py (2,C1) = §,Sp, = Sy = 34003 = 2pg «
By formula (16) s» = ha"2f4:; S21 = £1°P0y = tapos + ‘h: - fapa1 .
Therefore  Zos®bs = (#3° = 2t5)t01 = (P1"tos = papos + Paq = pabss)
= fapas = pobor = a1 o
F21(2,01) = 3(5:%S01 = 8,5,1 = 25,5, + 25,,]

1
= g[52"501 = S4801 — 255524 + 254,)

(2)  Sai®as®b,