Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/274

 Additional Theorems on Conjugate Functions.

187.] THEOREM IV. If $$x_{1} $$ and $$y_{1} $$, and also $$x_{2} $$ and $$y_{2} $$ are conjugate functions of $$x$$ and $$y$$, then, if

$X=x_{1}x_{2}-y_{1}y_{2} $ and $Y=x_{1}y_{2}+x_{2}y_{1} $

$$X$$ and $$T$$ will be conjugate functions of $$x$$ and $$y$$.

For

$X+\sqrt{-1}Y=\left(x_{1}+\sqrt{-1}y_{1}\right)\left(x_{2}+\sqrt{-1}y_{2}\right) $

THEOREM V. If $$\phi $$ be a solution of the equation

$\frac{d^{2}\phi}{dx^{2}}+\frac{d^{2}\phi}{dy^{2}}=0 $,|undefined

and if

$2R=\log\left(\left(\frac{d\phi}{dx}\right)^{2}+\left(\frac{d\phi}{dy}\right)^{2}\right) $, and $\Theta=\tan^{-1}\frac{\frac{d\phi}{dx}}{\frac{d\phi}{dy}} $,|undefined

$$R$$ and $$\Theta $$ will be conjugate functions of $$x$$ and $$y$$.

For $$R$$ and $$\Theta $$ are conjugate functions of $$\tfrac{d\phi}{dx} $$ and $$\tfrac{d\phi}{dy} $$, and these are conjugate functions of $$x$$ and $$y$$.

EXAMPLE I. – Inversion.

188.] As an example of the general method of transformation let us take the case of inversion in two dimensions.

If $$O$$ is a fixed point in a plane, and $$OA$$ a fixed direction, and if $$r=OP=ae^{\rho} $$, and $$\theta=AOP $$, and if $$x, y$$ are the rectangular coordinates of $$P$$ with respect to $$O$$,

{{MathForm2|(5)|$$\left.\begin{array}{lll} \rho=\log\frac{1}{a}\sqrt{x^{2}+y^{2}}, & & \theta=\tan^{-1}\frac{y}{x},\\ x=ae^{\rho}\cos\theta, & & y=ae^{\rho}\sin\theta, \end{array}\right\} $$}}

$$\rho $$ and $$\theta $$ are conjugate functions of $$x$$ and $$y$$.

If $$\rho'=n\rho $$ and $$\theta'=n\theta $$, $$\rho' $$ and $$\theta' $$ will be conjugate functions of $$\rho $$ and $$\theta $$. In the case in which $$n=-1 $$ we have

which is the case of ordinary inversion combined with turning the figure 180° round $$OA$$.

Inversion in Two Dimensions.

In this case if $$r$$ and $$r'$$ represent the distances of corresponding