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

76.] 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 $$R cos \epsilon$$ 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,

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 $$R cos \epsilon dS$$ 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

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

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