470 THE

TRANSCENDENCE

OF 7w

[October,

By equation (1), the elementary symmetric functions of

a,,

a2,

,

*? are

ra-

tional numbers. Hence the elementary symmetric functions

of the

quantities

(3)

are rational numbers. It follows that the quantities (3)

are

roots

of

(4)

02(x)

=

0,

an

algebraic equation with integral coef ficients. Similarly, the sums of

the

a's

taken three at a time are the nC3 roots of

(5)

03(X)

=

0

Proceeding thus, we obtain

(6) 04(X)

=

0,

05(X)

=

0,

,

.

.

v 0,(X)

=

0,

algebraic equations with integral coefficients, whose roots are the sums of the a's

taken 4, 5,

,

n at a time respectively. The product equation

(7) 61(X)02(X)

.

..(x)

=

0

has roots which are precisely the exponents in the expansion of (2).

The deletion of zero roots (if any) from equation (7) gives

(8)

0(X)

=

cxr

+

ClXr-l

+ + Cr 0,

whose

roots /i,

02

O

* r

are the

non-vanishing exponents

in

the

expansion

of

(2), and whose coefficients

are

integers.

Hence

(2) may be written

in the form

(9)

e#1

+

e#2

+

...

+

er

+

k=

0

where k is a positive integer.

We

define

( 10) ~~~~~~~~csxP1I{6(

x)

}

P~

(10)

f(x)

-

(sp- i)!

1

where s

=

rp -1, and p

is

a prime to be specified. Also we define

(11)

F(x)

=

f(x)

+

f(')(x)

+

f(2)(x)

+

.

. .

+

f(s+p+l)(x)

noting, with thanks to Hurwitz, that the derivative of e-xF(x) is

e-xf(x).

Hence we may write

e-xF(x)

-

eF

(0)

=

f

-

e-f(t)d

.

The

substitution t

=

-x produces

F(x)

-

exF(O)=

- X

e(1-7)x

f(rx)dT

.

Let x

range over

the

values

31,/2,

.2

,

* r

and add

the

resulting equations.

Using

(9).

we

obtain