Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/60

the values of $X,Y,Z$ being those at a point in the surface, and the integrations being extended over the whole surface.

If the surface is a closed one, then, when $y$ and $z$ are given, the coordinate $x$ must have an even number of values, since a line parallel to $x$ must enter and leave the enclosed space an equal number of times provided it meets the surface at all.

Let a point travelling from $x=-\infty$ to $x=+\infty$ first enter the space when $x=x_{1}$ then leave it when $x=x_{2}$, and so on; and let the values of X at these points be $X_{1},X_{2}$, &c., then

$\iint Xdydz=\iint \{(X_{2}-X_{1})+(X_{4}-X_{3})+\&c.+(X_{2n}-X_{2n-1})\}dy\,dz$

(2)

If $X$ is a quantity which is continuous, and has no infinite values between $x_{1}$ and $x_{2}$, then

$X_{2}-X_{1}=\int _{x_{1}}^{x_{2}}{\frac {dX}{dx}}dx$

(3)

where the integration is extended from the first to the second intersection, that is, along the first segment of $x$ which is within the closed surface. Taking into account all the segments which lie within the closed surface, we find

$\iint Xdydz=\iiint {\frac {dX}{dx}}dx\,dy\,dz$

(4)

the double integration being confined to the closed surface, but the triple integration being extended to the whole enclosed space. Hence, if $X,Y,Z$ are continuous and finite within a closed surface $S$, the total surface-integral of $R$ over that surface will be

$\iint R\cos \epsilon dS=\iiint ({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}})dx\,dy\,dz$

(5)

the triple integration being extended over the whole space within $S$.

Let us next suppose that $X,Y,Z$ are not continuous within the closed surface, but that at a certain surface $F(x,y,z)=0$ the values of $X,Y,Z$ alter abruptly from $X,Y,Z$ on the negative side of the surface to $X',Y',Z'$ on the positive side.

If this discontinuity occurs, say, between $x_{1}$ and $x_{2}$ , the value of $X_{2}-X_{1}$ will be

$\int _{x_{1}}^{x_{2}}{\frac {dX}{dx}}dx+(X'-X)$

(6)

where in the expression under the integral sign only the finite values of the derivative of $X$ are to be considered.

In this case therefore the total surface-integral of $R$ over the closed surface will be expressed by