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

20 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

If $$X$$ is a quantity which is continuous, and has no infinite values between $$x_1$$ and $$x_2$$, then

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

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

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

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