Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/117

this condition is shewn to be fulfilled by the electric forces with the most perfect accuracy. Hence the law of electric force is verified to a corresponding degree of accuracy.

Surface-Integral of Electric Induction, and Electric Displacement through a Surface.

75.] Let $R$ be the resultant force at any point of the surface, and $\epsilon$ the angle which R makes with the normal drawn towards the positive side of the surface, then $Rcos\epsilon$ is the component of the force normal to the surface, and if $dS$ is the element of the surface, the electric displacement through $dS$ will be, by Art. 68,

${\frac {1}{4\pi }}\,KR\,cos\epsilon \,dS$

Since we do not at present consider any dielectric except air, $K=1$ .

We may, however, avoid introducing at this stage the theory of electric displacement, by calling $Rcos\epsilon dS$ the Induction through the element $dS$. This quantity is well known in mathematical physics, but the name of induction is borrowed from Faraday. The surface-integral of induction is

$\iint Rcos\epsilon dS$;

and it appears by Art. 21, that if $X,Y,Z$ are the components of $R$, and if these quantities are continuous within a region bounded by a closed surface $S$, the induction reckoned from within outwards is

$\iint Rcos\epsilon dS=\iiint \left({\frac {}{}}+{\frac {}{}}+{\frac {}{}}\right)dx\,dy\,dz$,

the integration being extended through the whole space within the surface.

Induction through a Finite Closed Surface due to a Single Centre of Force.

76.] Let a quantity e of electricity be supposed to be placed at a point $0$, and let $r$ be the distance of any point $P$ from $0$, the force at that point is $R={\frac {e}{r^{2}}}$ in the direction $OP$.

Let a line be drawn from $O$ in any direction to an infinite distance. If $O$ is without the closed surface this line will either not cut the surface at all, or it will issue from the surface as many times as it enters. If $O$ is within the surface the line must first