_11-
are functions of the s's defined on page 3. Next we shall set
1 = ¢, and expand this according to powers of w. Each power of
v is to be arranged according to powers of x, and then e¢¥ is to be
arranged according to powers of x, the coefficients Cy being func-
tions of the s's,

In order tc begin the work outlined abcve, we gust have
each v,(x) converge and |vy(x)] £1 but vy(x) =% - 1. We shall suppose
that these conditions are satisfied.

Consider now the logarithm of the ith factor of II;

(__1)1l—1

(4)  legll+vi(x)] = vimgu+dv®=croers gy ® woens

By using the multinowial theorem we obtajn
s s GIE

—— U§ =
n ”

(-)™? n!

e & e (Y A ernr Uy

 

(fixa + 6‘18 NS lt!l 4 veees )11

(-1)"
(5) = Z(—-%-,-;—ai"bt"‘” W,
qe

where for abbreviation we define Ji! = jlkl***r!, w=w~1, and
Jj*+ B+ **+ra=h, and where the summation is taken only for

the non-negative values cf j, &k, °***, r, which satisfy the equation
jtk+ et tr=n_ We see that j, &, ***, r, are not all zero at
once, fe note that J1!% = ji%kl¥eee ,g%

Since (4) is a convergent power series, it converges abso-
lutely and uniformly® for |v;l<1~-¢, where ¢ is an arbitrary
positive puwber less than 1, Let R, dencte the radius of a circle
such that Jvi(x)l <1~¢ when |x| < Ry, and i =1, 2, 3, *=*+ , 80 that
the series (4) is absolutely and uniformly convergent when lx] <« Ry
fle now impose the condition on the elements of the array (1) that
they be chosen so that R, shall be ¢reater than zero, Since vwy(x)

e e e e e e el e T 3 8 i e it e

’ Bromwich, sec, 50. Smail, seo, 165, 169. Tannery, sec. 184,