Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/121

78.] length $$v$$, till it meets the surface $$F = a + h$$, then the value of $$F$$ at the extremity of the normal is

The value of $$V$$ at the same point is

Since the first derivatives of $$V$$ continue always finite, the second side of the equation vanishes when $$h$$ is diminished without limit,  and therefore if $$V_2$$ and $$V_1$$ denote the values of $$V$$ on the outside  and inside of an electrified surface at the point $$x, y, z,$$

If $$x +dx$$, $$y + dy$$, $$z + dz$$ be the coordinates of another point on the electrified surface, $$F=a$$ and $$F=a+h$$ at this point also ; whence

and when $$dx$$, $$dy$$, $$dz$$ vanish, we find the conditions

{{numb form }}
 * $$\left .{\begin{matrix}\dfrac{dV_2}{dx}-\dfrac{dV_1}{dx}=Cl, \\ \\ \dfrac{dV_2}{dy}-\dfrac{dV_1}{dy}=Cm\\ \\ \dfrac{dV_2}{dz}- \dfrac{dV_1}{dz}=Cn \end{matrix}} \right \}$$
 * (12)

where $$C$$ is a quantity to be determined.

Next, let us consider the variation of $$F$$ and $$\frac{dV}{dx}$$ along the ordinate parallel to $$x$$ between the surfaces $$F= a$$ and $$F = a + h$$.

Hence, at the second surface, where $$F=a + h$$, and $$V$$ becomes $$V_2$$,