APFENDIX.

 

Note concerning the weights.

When the number of terms of each factor is finite,

some of
the numbers @, B, y, -*-- may be equal (a = Q). Requiring each of

the a's, b's, ¢'s, etc. to keep its identity without being combined
with some other element because of the equalities among the expo~
nents in vy(x), we see that pjr...r and sjr...r are as defined on
pages 2 and 3, and each is of weight h=ja+ B + ** 4+ A, Theorem I
does not depend on the weights and therefore holds true for this
case. There is nothing in the proof of Theorem II which is affected
by having the weights equal, and therefore Theorem II holds true.
Froceeding through the proof of Theorem III, we see that the re-
arrangement of terms to obtain formula (5) may be performed since
v¢ is a polynomial and v;" therefore satisfies the condition of
atsclute convergence. Therefore the conditions of the Weierstrass _ 5
theorem are satisfied and we can select the terms of ug, as before,
Now the series %u,h e log [1+ v4] is absclutely convergent, and
Z‘log[l +vil =1leg IT is by hypothesis uniformly convergent, so that

by the Weierstrass thecrem 53 uy R R e and
: h ih

0

 

th hoi
o ~1)" m! s o (<1)™ m! e ()"
Zap =21 gt e =Rt ey = g EER

where ja+ B+ +ri=h, and m=j+k+ v 4pr-1 , and previded
the s's converge. This interchange of order of summation may be
made because the nurber of terms in ugp is finite when there is not
an infirity of solutions to the equaticn ju + B+ +rA=h, There
could not be an infinity of sclutioms with only a finite number of
weights a, B, v, ==+, A\, a% 0. We have now obtained formula (7)
as before, and the reraining portion of the proof fellows without
changes, and we find that Theorems ITla, IIIb, and IV as stated will
apply to this case.