-8-

will be called the weights of the elements ay, by, c,, <tee ., The
weight will always be denoted Ly the Greek letten corresponding to
the letter representing the element. The weight of a term which is
the product of several elements of the sare or different sequences
is the sum of the weights of the elemepts waking up the term, For
example, the weight of each term of Se i3 3a, of sz, is 5¢ + 28,
and in general, the weight of Sjke++q 1is dencted by 4, where
h= g0+ k4 4o . The weight of ¢, is alse 3o, of ¢,, is
5a + 28, and in gereral, tre weight of #yhe0eq is b . If 21l the
teres of a symnetric furction have the same weight, the weight of
the functicn is defined tc be the weight of any one of its terms,
The idea of weights is nct essential to the theory, but it sicpli~
fies the discussion of the relations between symmetric functions,

In order tc develop the formulas of this section, let the
elements of the array (1) satisfy the following conditions: all
the ¢'s converge, each vy(x) ccnverges and is not equal to - 1,
and ¥ lv,(x)]| is uniferaly convergent fer a certain range of valuves
of x. From these conditions we see that I is ahsclutely—unifcrmly
convergent. Since II is unchapged in value when the rows of (19
are¢ permuted, it is a symmetric function of the elements, Writing

cut II we have

[l " O;Za + b’XB # caxY Frisieives g l,x)‘ ¥ --.v.]
x [1 + a,xa + b,:e + c,xY Fovmng l,x)‘ 3} Sormisk )
x [1 4 agx® 4 bgab + cgxl 4 reveey 151A+ vieisaviy

x-----v.-..-.--..-n---.o
-

When the wultiplication is perforzed, any term of the resulting
expansion is a product which can bLe formed by taking exactly one
term from each of the factors of II. Taking 1 from each factor,
the first term of the expansion is 1, Taking 1 from each factor

except one, we get all the terrs

Saga® & IBeta zheal sitrne i S uina o o Y ook