_28_
The most general I functions in terms of

the s's, and in terms of the #'s.

In order tc complete the problem of expressing any
sympetric I function in terms of the #'s or the s's, it will be
convenient to use the following'notation:

(agdgeg*ec) = (abeer*), ,
Py(1) = Jag , Fy4(1,2) = Jagas® , F111(1,2,3) = Z"i“da‘g‘ ’
F21(2,01) = Za,%as™, , F22(31,12) = 2(a®); (a®), (ab*®), (a%), ,
F111(2,01,12) = Zfldabj(ab')k , F112(2,01,12) = Zflqzbj(ab‘)k (ab')l .
The numbers between commas in the parentheses are the exponents
of a set of letters (abc:") in the order given. Each set between
coemas is used like an "element" in the sum P. The subscripts refer
to these "elements" with the same meaning that they have in con-
nection with the ¢'s. Suppose for example in F,, above, we let
(@®)i=T¢, and (@*);=7'. Aith this substitution, Fyy = 31,17
that is, the sum of terms made up of two elements of the first
kind and two of the second kind, each taken from a different row. ®®
Compare with #35 = Sagasbyb; .

We define a set of S's corresponding to the P's:
S21(2,01) = (a5 *)2(8¢) , S22(81,12) = 3(a), *(ab?),? ,
$112(2,01,12) = 3(a®)4(b)4 (ab®), 2 .
It is evident that these S's reduce immediately to the s's, and
that is what makes ther valuable,

Now if we expand into a series the product
TI01 + @512 4000y o 4 (Fomfoefoan.,y o (15505 o0 nny 3y ey

in which we suppose that the quantities in parentheses satisfy the

-—;—_——MM

2 The T's are used hera in the same sense in which Cullis uses them to denode
the 'type' of the sum. Maomahon uses the psrent®esis notetion. My notation is a
combination of the two, (and I think it is easier to comprehend and easier to
work with),