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

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

and if

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

$R$ 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 $O$,

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

(5)

$\rho$ 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

$r'={\frac {a^{2}}{r}},\ \mathrm {and} \ \theta '=-\theta$,

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