-43-

If some of the weights are equal, some of the p's and s's
appear in C's of lower weight than they do if the weights are
1, 2, 3, **- . A ccnvenient way of computing the C's in terms of
the s's is first to find the C's ip terms of the #'s, and then
substitute for each ¢ in terms of the s's, since the relations
between the ¢'s and s's are independent of the weights,

If o= Bim roeer =X =1, then **
Ci=ts+toa+tteost **°* =83+ Sca+ Seoa t *°°°°3
C2a= (#a + tea + fooe + ***) t (P12 + ta01 + boaa t ***°)

= F(ss% 4 83 4 ) = (554 sca + ***) = (531 4 S101 % ***) + (84503 + 335605 **)

Ca=(fa+tos+ ***) + (Fan+ faa + **7) + (F211 4 fy01a + foraa + ****)

If the product Il has only a finite number of factors and
each factor is a polynomial, converdence does not need to be con-
sidered and all the cperations performed in deriving the formulas
Leccme algetraic cperaticns, All the thecreme and feormulas are
valid in this special case, Lut there do nct need tc be any condi-

tions whatever on the elements of the array (1) or the functions vi.

Nete on the choice of notation.

The nctation used in this paper cannot be called "standard”
for if one readson the subject of sympetric functicns, he will see
that no twec writers use exactly the same notation; that is, there
is no standard, Careful consideration has Leen given to the choice
of notation used, an effort being made to select symbols which agree
with as wany writers as possible, Only a few writers have considered
more than one sequence of elements, but nevertheless, there are
differences in the use of the subscripts attached teo the letters
denoting functions. For exawmple, compare Junker with Macmahop, 1 do
not maintain that my notaiioh is the best that can be devised, but
I believe it is systepatic and readatle,

3% See Cullis, Vol.III, Chap. 38, end Macmahon, Vol, II, Seo. XI,