Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/121

length $v$, till it meets the surface $F=a+h$, then the value of $F$ at the extremity of the normal is

$a+v\left(l{\frac {dF}{dx}}+m{\frac {dF}{dy}}+n{\frac {dF}{dz}}\right)$+&c.,

(5)

or

$a+h=a+vR+\,\!$&c.

(6)

The value of $V$ at the same point is

$V_{2}=V_{1}+v\left(l{\frac {dV'}{dx}}+m{\frac {dV'}{dy}}+n{\frac {dV'}{dz}}\right)+$ &c,

(7)

or

$V_{2}-V_{1}={\frac {h}{R}}{\frac {dV'}{dv}}+$ &c.

(8)

Since the first derivatives of $V$ continue always finite, the second side of the equation vanishes when $h$ is diminished without limit, and therefore if $V_{2}$ and $V_{1}$ denote the values of $V$ on the outside and inside of an electrified surface at the point $x,y,z,$

$V_{1}\,=\,V_{2}$

(9)

If $x+dx$, $y+dy$, $z+dz$ be the coordinates of another point on the electrified surface, $F=a$ and $F=a+h$ at this point also ; whence

$0={\frac {dF}{dx}}dx+{\frac {dF}{dy}}dy+{\frac {dF}{dz}}dz+$ &c.,

(10)

$0=\left({\frac {dV_{2}}{dx}}-{\frac {dV_{1}}{dx}}\right)dx+\left({\frac {dV_{2}}{dy}}-{\frac {dV_{1}}{dy}}\right)dy+\left({\frac {dV_{2}}{dz}}-{\frac {dV_{1}}{dz}}\right)dz+$ &c.;

(11)

and when $dx$, $dy$, $dz$ vanish, we find the conditions

$\left.{\begin{matrix}{\dfrac {dV_{2}}{dx}}-{\dfrac {dV_{1}}{dx}}=Cl,\\\\{\dfrac {dV_{2}}{dy}}-{\dfrac {dV_{1}}{dy}}=Cm\\\\{\dfrac {dV_{2}}{dz}}-{\dfrac {dV_{1}}{dz}}=Cn\end{matrix}}\right\}$

(12)

where $C$ is a quantity to be determined.

Next, let us consider the variation of $F$ and ${\frac {dV}{dx}}$ along the ordinate parallel to $x$ between the surfaces $F=a$ and $F=a+h$.

We have

$F=a+{\frac {dF}{dx}}dx+{1 \over 2}{\frac {d^{2}F}{dx^{2}}}(dx)^{2}+$ &c,

(13)

and

${\frac {dV}{dx}}={\frac {dV'}{dx}}+{\frac {d^{2}V'}{dx^{2}}}dx+{\tfrac {1}{2}}{\frac {d^{3}V'}{dx^{3}}}(dx)^{2}+$ &c.,

(14)

Hence, at the second surface, where $F=a+h$, and $V$ becomes $V_{2}$,

${\frac {dV_{2}}{dx}}={\frac {dV_{1}}{dx}}+{\frac {d^{2}V'}{dx^{2}}}dx+$ &c.;

(15)