_19—
for the C's for the set 1, 8, 4, **-*, instead of setting B = C as
we might be inclined to do, or B = 3, y = 4, ete., we set all the
b's equal to zero. This causes all the #'s and s's to ke zero {n
which the second subscript shows that some of the b's are factors
of each term; that is, if the second subseript is different froe
zero, the p or s is zero. Therefore the expressions given in (12)
way be used, and they may be shertened by omitting every term in
which there is a p or s whose second subscript is not zero, This
also allows us to use unchanged the formula for the weights of the
functions, namely h=ja+ 4+ ****+r\. A similar discussion coyld
te given for any set of weights satisfying the definition of vi(x),
so that we may take the weights to be 1, 3, 3, =**v, by defining

soze of the elements cf the array (1) to be zero if necessary,

Expressions for the p's in terms of the s's,

and for the s's in terms cf the p's,

We shall now show how to obtain the expression for Pihoser

in terms of the s's.

Form the array of elements (1') using the sequences '{f a4},
1R Yelsiesren, (lkl¢}, ***°, where Yy is a variable parameter
having the same value for all the a's at the same time, Y3 one
having the same value for all the d's at the same time, etc., Let
the elements of the array (1') be such that when = % m e T it
the conditions of Theorew IV are satisfied. Using the elements (1')
form the v's as defined on page 7, and form the infinite prodyct
II[(1 + vi(x)] . By Thecrem IV we now cbtain the C's as functions of
Tyae, Ighy, ***°, By, ***+ -, letting p' and s' denote the p's
and s's for (1'), we obtain?®

P akeeer = Y.,"’Yg'flfl’{fi.ih-..r and  s'ype..y = Ya"l’e'""]{s“...' .

ettt ettt e eagers

25 Broswioh, sec. 18, Smail, sec. n4.