Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/70

 are all integers divisible by $$p$$. This follows from the fact that $$\textstyle \sum\limits^{m=n}_{m=1} (C\beta_m)^r$$ is expressible in terms of those symmetrical functions which

consist of the sums of products of the numbers $$C\beta_1, C\beta_2, \ldots$$; and these expressions have integral values.

(4) Let $$K_p$$ denote the integer

which may be written in the form

In virtue of what has been established in (3) as to the values of

we see that $$K_p A$$ is not a multiple of $$p$$.

"We examine the form to which the equation

is reduced by multiplying all the terms by $$K_p$$. We have

r=np+p-I

r=p-l

Q r+i O r+2

"*" T "*" 7 ~ / ^T +

The modulus of the sum of the series

does not exceed

and this is less than $$e^{|\beta_m|}$$; hence we have

where $$\theta_r$$ is some number whose modulus is between 0 and 1.

The modulus of $$\textstyle \sum\limits^{r=np+p-1}_{r=p-1} \theta_r |c_r {\beta_m}^r| e^{|\beta_m|}$$ is less than $$\textstyle e^{|\beta_m|} \sum\limits^{r=np+p-1}_{r=p-1} \theta_r |c_r {\beta_m}^r|$$,