Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/69

The general value of $\beta$ consists of the sum of $r$ of the letters $\alpha _{1},\alpha _{2},\ldots \alpha _{s}$; and we consider those values of $\beta$ that correspond to a fixed value of $r$ to belong to one set. It is clear that a symmetrical function of those letters $\beta$ which belong to one and the same set is expressible as a symmetrical function of $\alpha _{1},\alpha _{2},\ldots \alpha _{s}$; therefore a symmetrical function of the products $C\beta$, where all the $\beta$'s belong to one and the same set, is in virtue of what has been established in (1) an integer. Applying the above lemma to all the $n$ numbers $C\beta$, we see that the symmetrical products formed by all the numbers $C\beta$ are integral, or zero. We have supposed those of the numbers $\beta$ which vanish to be suppressed and the corresponding exponentials to be absorbed in the integer $A$; whether this is done before or after the symmetrical functions of $C\beta$ are formed makes no difference, so that the above reasoning applies to the numbers $C\beta$ when those of them which vanish are removed.

(3) Let $p$ be a prime number greater than all the numbers $A$, $n$, $C$ $|C^{n}\beta _{1}\beta _{2}\ldots \beta _{n}|$; and let

$\phi (x)={\frac {x^{p-1}}{(p-1)!}}C^{np+p-1}\{(x-\beta _{1})(x-\beta _{2})\ldots (x-\beta _{n})\}^{p}$.

We observe that $\phi (x)$ is of the form

${\frac {(Cx)^{p-1}}{(p-1)!}}[(Cx)^{n}-q_{1}(Cx)^{n-1}+q_{2}(Cx)^{n-2}-\ldots +(-1)^{n}q_{n}]^{p}$,

where $q_{1},q_{2},\ldots q_{n}$ are integers. The function $\phi (x)$ may be expressed in the form

$\phi (x)=c_{p-1}x^{p-1}+c_{p}x^{p}+\ldots +c_{np+p-1}x^{np+p-1}$,

where $c_{p-1}(p-1)!,c_{p}p!,\ldots$ are integral.

We see that $\phi ^{p-1}(0)=(-1)^{np}C^{p-1}q_{n}^{p}$, which is an integer not divisible by $p$.

Also $\phi ^{p}(0)$ is the value when $x=0$ of

$pC^{p-1}{\frac {d}{dx}}[(Cx)^{n}-q_{1}(Cx)^{n-1}+\ldots ]^{p}$

and is clearly an integer divisible by $p$. We see also that

$\phi ^{(p+1)}(0),\phi ^{(p+2)}(0),\ldots \phi ^{np+p-1}(0)$

are all multiples of $p$.

Further if $m\leq n$, $\phi (\beta _{m}),\phi '(\beta _{m}),\ldots \phi ^{(p-1)}(\beta _{m})$ all vanish, and

$\textstyle \sum \limits _{m=1}^{m=n}\phi ^{(p)}(\beta _{m}),\sum \limits _{m=1}^{m=n}\phi ^{(p+1)}(\beta _{m}),\ldots \sum \limits _{m=1}^{m=n}\phi ^{(mp+p-1)}(\beta _{m})$