_21-
35:3¢ the summation is taken for the values of the indices which
'satisfy the squations Inyjq = j, Sngkg =k, ***2*, Ingry = v,
and where my=jq+ ki+ ' +ry-1,

In equations (12) the terms are grouped in parentheses so
that the first, second, thirq, etc. group represents the first,
second, third, etc. ¢ respectively. The first group in each equs—'
tion is the well know expression for #s5 in terms of s's, which is
a special case of (13) where by = ¢4 = ***** = (,

In order to call special attention to the fact, we state
the following

CorOLLARY V.1. The regresentation of $54...¢ ii terms of

the s's is imdependent of the meights a, B, y, ===

Another interesting point to notice is that the coefficient

of the product of s's in (13) is symmetric in the a's, n's, and J's,
so that if j, %, ***, #, are permuted, the coefficients of the
expression in s's are unchanged. Only the subscripts are pernuted,
For example, fa1 = #(s53"sox = 234811 + 285 = Sg505) ,

Pre = (s1802" - 28503533 + 28, - $1803) ,

toiz = (s013001" = 250015011 + 28015 = S015002) -

CoROLLARY V.2, Let j', k', v==«, r', be a permutation of
Js» k, ****, r- . The extressions for Pihesor and for g,1.0 1 have
the same coefficients, and the indices of one expression may be
obtained from the other by persutation,

We shall now derive some relations between the s's and C's,
and expressions for the s's in terms of the ¢'s. Suppose the
conditions cf Theorem IV are satisfied, so that ¢

H =14 0Cax +7Caz® + Cax® 4 eoveee,
Then  w=log Il = (1 ~1) - §(IT = 1)* 4 §(I = 1)%-vee-

(_l)fl-l

= Z (Cgl + ng? + C.x‘a 4 seeve )n 7

The series for II = 1 is absolutely convergent when |x| <K, , and