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

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

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,

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

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

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

{{numb form }}
 * $$\iint \left \{ \frac{dX}{dz}(\frac{dz}{d\alpha} \frac{dx}{d\beta}- \frac{dz}{d\beta}\frac{dx}{d\alpha})- \frac{dX}{dy}(\frac{dx}{d\alpha} \frac{dy}{d\beta}- \frac{dx}{d\beta} \frac{dy}{d\alpha} \right \rbrace d\beta \, d\alpha$$;
 * (6)

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

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

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