Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/410

equal to that due to a system of charges placed in air at certain points along the normal to the plate through $S.$

Make

$AI=SA,\quad BI_{1}=SB,\quad AJ_{1}=I_{1}A\quad BI_{2}=J_{1}B,\quad AJ_{2}=I_{2}A,\;\mathrm {\&c.;}$

then we have two series of points at distances from each other equal to twice the thickness of the plate.

318.] The potential in the first medium at any point $P$ is equal to

${\frac {E}{PS}}+{\frac {I}{PI}}+{\frac {I_{1}}{PI_{1}}}+{\frac {I_{2}}{PI_{2}}}+\;\mathrm {\&c.}$

(8)

that at a point $P'$ in the second

${\begin{array}{l}{\frac {E'}{P'S}}+{\frac {I'}{P'I}}+{\frac {I_{1}'}{P'I_{1}}}+{\frac {I_{2}'}{P'I_{2}}}+\;\mathrm {\&c.} \\\;\;\;+{\frac {J_{1}'}{P'J_{1}}}+{\frac {J_{2}'}{P'J_{2}}}+\;\;\mathrm {\&c.,} \end{array}}$

(9)

and that at a point $P''$ in the third

${\frac {E''}{P''S}}+{\frac {J_{1}}{P''J_{1}}}+{\frac {J_{2}}{P''J_{2}}}+\;\mathrm {\&c.,}$

(10)

where $I,\,I',$ &c. represent the imaginary charges placed at the points $I,$ &c., and the accents denote that the potential is to be taken within the plate.

Then, by the last Article, for the surface through $A$ we have,

$I={\frac {k_{2}-k_{1}}{k_{2}+k_{1}}}E,\quad \quad E'={\frac {2k_{2}}{k_{2}+k_{1}}}E.$

(11)

For the surface through $B$ we find

$I_{1}'={\frac {k_{3}-k_{2}}{k_{3}+k_{2}}}E',\quad \quad E'={\frac {2k_{3}}{k_{2}+k_{1}}}E'.$

(12)

Similarly for the surface through $A$ again,

$J_{1}'={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}I_{1}',\quad \quad I_{1}={\frac {2k_{1}}{k_{1}+k_{2}}}I_{1}',$

(13)

and for the surface through $B,$

$J_{2}'={\frac {k_{3}-k_{2}}{k_{3}+k_{2}}}J_{1}',\quad \quad J_{1}={\frac {2k_{3}}{k_{3}+k_{2}}}J_{1}'.$

(14)

If we make

$\rho ={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}\quad \mathrm {and} \quad \rho '={\frac {k_{3}-k_{2}}{k_{3}+k_{2}}},$

we find for the potential in the first medium,

${\begin{array}{rl}V={\frac {E}{PS}}-\rho {\frac {E}{PI}}+(1-\rho ^{2})\rho '{\frac {E}{PI_{1}}}&+\rho '(1-\rho ^{2})\rho \rho '{\frac {E}{PI_{2}}}+\;\mathrm {\&c.} \\&\quad +\rho '(1-\rho ^{2})(\rho \rho ')^{n-1}{\frac {E}{PI_{n}}}.\end{array}}$

(15)