Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/66

of the surface dS reckoned in the positive direction. Then the value of the surface-integral of ${\mathfrak {B}}$ may be written

$\iint (l\xi +m\eta +n\zeta )dS$

(2)

In order to form a definite idea of the meaning of the element $dS$, we shall suppose that the values of the coordinates $x,y,z$ for every point of the surface are given as functions of two inde pendent variables $\alpha$ and $\beta$. If $\beta$ is constant and $\alpha$ varies, the point $(x,y,z)$ will describe a curve on the surface, and if a series of values is given to $\beta$, a series of such curves will be traced, all lying on the surface $S$. In the same way, by giving a series of constant values to $\alpha$, a second series of curves may be traced, cutting the first series, and dividing the whole surface into elementary portions, any one of which may be taken as the element $dS$.

The projection of this element on the plane of $y,z$ is, by the ordinary formula,

$ldS=({\frac {dy}{d\alpha }}{\frac {dz}{d\beta }}-{\frac {dy}{d\beta }}{\frac {dz}{d\alpha }})d\beta \,d\alpha$.

(3)

The expressions for $mdS$ and $ndS$ are obtained from this by substituting $x,y,z$ in cyclic order.

The surface-integral which we have to find is

$\iint (l\xi +m\eta +n\zeta )dS$;

(4)

or, substituting the values of $\xi ,\eta ,\zeta$ in terms of $X,Y,Z$,

$\iint (m{\frac {dX}{dz}}-n{\frac {dX}{dy}}+n{\frac {dY}{dx}}-l{\frac {dY}{dz}}+l{\frac {dZ}{dy}}-m{\frac {dZ}{dx}})dS$;

(5)

The part of this which depends on $X$ may be written

$\iint \left\{{\frac {dX}{dz}}({\frac {dz}{d\alpha }}{\frac {dx}{d\beta }}-{\frac {dz}{d\beta }}{\frac {dx}{d\alpha }})-{\frac {dX}{dy}}({\frac {dx}{d\alpha }}{\frac {dy}{d\beta }}-{\frac {dx}{d\beta }}{\frac {dy}{d\alpha }}\right\rbrace d\beta \,d\alpha$;

(6)

adding and subtracting ${\frac {dX}{dx}}{\frac {dx}{d\alpha }}{\frac {dx}{d\beta }}$ this becomes

$\iint \left\{{\frac {dx}{d\beta }}({\frac {dX}{dx}}{\frac {dx}{d\alpha }}+{\frac {dX}{dy}}{\frac {dy}{d\alpha }}+{\frac {dX}{dz}}{\frac {dz}{d\alpha }})\right.$

$\left.-{\frac {dx}{d\alpha }}({\frac {dX}{dx}}{\frac {dx}{d\beta }}+{\frac {dX}{dy}}{\frac {dx}{d\beta }}+{\frac {dX}{dz}}{\frac {dz}{d\beta }})\right\rbrace d\beta \,d\alpha$;

(7)

$=\iint ({\frac {dX}{d\alpha }}{\frac {dx}{d\beta }}-{\frac {dX}{d\beta }}{\frac {dx}{d\alpha }})d\beta \,d\alpha$.

(8)

As we have made no assumption as to the form of the functions $\alpha$ and $\beta$, we may assume that $\alpha$ is a function of $X$, or, in other words, that the curves for which $\alpha$ is constant are those for which