_14_
and impose the condition on the array (1) that the elements bé
such that Zlog[l + vy(x)] converge uniformly. This is equivalent to
the condition that II converge uniformly. ** Using the definition
of w, and the definitions of the s's, we see that provided all. of
the s's converge :

-1 =1)" m¢
!lqh*):?.( ) b“.o.k ,zz( ) b{""'hr 22( Jz!"is't"‘r 3

 

 

where the unmarked ¥ denctes summation for those values of the
indices which satisfy the equation ja+ #8 + **** 4 rA = h, and where
m=j+k+***+r-1, Since uy has a finite number of terms, as
shown. before, the change of order of the repeated sums is permis—~
sible if the s's converge. The convergence of the s's does not
follow from the convergence of zu", « We must now impose the con-
dition that the s's converge.

New we define

" m
(7) L ] 2—"’—“'_ Sdkheser = %t Evib »

where the summation is taken for the values of the indices which
satisfy the equation ja+ M+ ***4+rA=h. With this definition
we can write

u= Zlog[l + vy(x)] = Eu,, ’

and the last series converges absolutely, since it is a pewer senes'

M‘

Sj 1 u u® W0
Sipce I = e =.‘1+u+§,- +'3'T + +a_+

.

2

it is necessary tc find expressions fer these terms.
Let Vpgo Ypge *°°% denote any of the terws cf the series
u = Juy, where hy, hy, ****, are distinct and denote a set of num—

bers selected from 1, 2, 3, «-+ , Ry using the multinomial theorer

< 1 Hive DRI EN P
(8) 5’? e n!a!-q-pl “h: uhg uh' ’

e e Sl

e Smajl, seo, 257.

~4