Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/280

The points for which $\psi$ is constant lie in the hyperbola whose axes are $2\cos \psi$ and $2\sin \psi$.

On the axis of $x$, between $x=-1$ and $x=+1$,

$\phi =0,\ \psi =\cos ^{-1}x$

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On the axis of $x$, beyond these limits on either side, we have

${\begin{array}{lllll}x>1,&&\psi =0,&&\phi =\log \left(x+{\sqrt {x^{2}-1}}\right),\\x<-1,&&\psi =\pi ,&&\phi =\log \left({\sqrt {x^{2}-1}}-x\right),\end{array}}$

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Hence, if $\phi$ is the potential function, and $\psi$ the function of flow, we have the case of electricity flowing from the negative to the positive side of the axis of $x$ through the space between the points -1 and +1 , the parts of the axis beyond these limits being impervious to electricity.

Since, in this case, the axis of $y$ is a line of flow, we may suppose it also impervious to electricity.

We may also consider the ellipses to be sections of the equipotential surfaces due to an indefinitely long flat conductor of breadth 2, charged with half a unit of electricity per unit of length.

If we make $\psi$ the potential function, and $\phi$ the function of flow, the case becomes that of an infinite plane from which a strip of breadth 2 has been cut away and the plane on one side charged to potential $\pi$ while the other remains at zero.

These cases may be considered as particular cases of the quadric surfaces treated of in Chapter X. The forms of the curves are given in Fig. X.

Fig. X.

EXAMPLE VI. Fig. XI.

193.] Let us next consider $x'$ and $y'$ as functions of $x$ and $y$, where

$x'=b\log {\sqrt {x^{2}+y^{2}}},\ y'=b\tan ^{-1}{\frac {y}{x}}$

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$x'$ and $y'$ will be also conjugate functions of $\phi$ and $\psi$.

The curves resulting from the transformation of Fig. X with respect to these new coordinates are given in Fig. XI.

Fig. XI.

If $x'$ and $y'$ are rectangular coordinates, then the properties of the axis of $x$ in the first figure will belong to a series of lines parallel to $x'$ in the second figure for which $y'=bn'\pi$, where $n'$ is any integer.

The positive values of $x'$ on these lines will correspond to values of $x$ greater than unity, for which, as we have already seen,

$\psi =n\pi ,\ \phi =\log \left(x+{\sqrt {x^{2}-1}}\right)=\log \left(e^{\frac {x'}{b}}+{\sqrt {e^{\frac {2x'}{b}}-1}}\right)$

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