_23_

This equality is true whether @BY ey e NS A
or other numbers, but if they are other numbers the coefficients in
the second of the two equations given above the theorem must be
changed to correspond. The coefficients in terms of the C's will
remain unchanged. In the special case where b; = c; = seete =0, all of
the #'s and s's are zero except p; and s; . Then we obtain the
well known formulas

s3=$1, Sa=¢," -20, $g= ha'3fi:?a + 3ps , ".,“.“ .

To obtain w and uy in terms of #'s, we substitute formula
(3') into (14). We use (3') because the C's are not independent of
GaPy YomEeRs DS Tace 0 s Zfi[gé‘"il, where r+ s+t tt=w,

by the multinomial theorenm

Cw: » 211'1""- J {’[ g ]}j’ """{ :(;1 o2t = ]}JH

where Yt oate 4 tfl s for n = 1:2:"', il and where Ji j: + °°° 4 ji, =3

Then jara + ***+ jata + oo 4 .71-1"

iy SRR o ]i;ti,

> CJawy s eee +j1'1w, = jwy .,

=)™ m!

Now the product 3

C," C,k coee CK"
_tx;]}]‘..... {fi[% o %;]}j":

®secscevse

e
13

.t A Y Mg
1} fol ot 40}
o u "ix!
([jarx 4 e0e 4 j:tg] 4 vesee g [jilri 4 e g4 jiiti,]) 4 scesss
+ ([":'x 4 0 4 "lt ] 4 cesee 4 ["IK'*K 4 cee g "!'ytl;;(])
(j,'l“‘ eesee 4 jij‘ 1) 4 vreenee + (fl,'K'{' "‘.'+fl’.K.K)
A-7) + @°k) +- *o000c 4 (Kom) = b

from the condition on (14) . Since each C contains only a finite