-6~

Now $4 = a,J + ngl + a" 4 *eess
and 8, = blh + b,“ + b.‘ 4 veces

e s
so that Ssk= a,zb,s + a,’b,z + ag?d g% 4 o,

Multiplying the last lime term by term by a third s, and that
result by another s, etc,, we obtain $ikeeep ', Which is absolutely
convergent for each choice of j, k, ****, r, Therefore all the
s's are absolutely coovergent if s,, sq1, $e0s, **** are,
The defipitions. of the #'s and s's show that $3, foa,
Poos, "°°, are equal respectively to S5, o3, Sgoa, ****, s0 thai
when the functions of either of these sets are symmeiric, so are
the functions of the other set.
Let 25 = a5 4+ a;, + a4 “t°®* converge absolutely, and
write s = ayay + 0,05 + aj04 + agag 4 e

+ aga,

5
+

‘+ agag + agas + agag + e
+ agag 4 *ccee
+

tesensvessace

« o
200000 e = za‘ E al'
=1 2441

Each line of this series is absclutely convergent, for it
can be written aqg ;‘,a, » and %‘ lagl S § laj| , which is convergent
41 141 3 &
because £y is absolutely convergent. Let A denote Tlayl . Now
1
replace each factor of each term by its absolute value, and we
o« « o L3 2 R ?
have zla¢lz las| = Zla-’lZIGJ|=A » from which ¢, & A%, and ¢,
converges absqlutely. Slmxlarly we can write |[gg] = |Za¢ 2 aJ z a.l
i
s Elailz’aJlZIOkl A*, aud therefore ¢, converges absolutely. We

can show in the same manner that every ¢4 converges absolutely for

every j. Now write

$11 = ; G3bg + a3y 4 asby 4 ceves
agbs * 4+ asby + agbe + ccvee
aghy + aghy * 4+ agbg 4 *eves
asba + asba + asds * 4 seeee

PeP0r0svr0ecrrrerrec et ennng e

ek e

»