Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/201

 and if a point in all respects equal and of opposite sign be placed at the origin, the potential due to the pair of points will be

$$\begin{array}{ll} V & =Mf\left\{ (x-lh),\ (y-mh),\ (z-nh)\right\} -Mf(x,y,z),\\ \\ & =-Mh\frac{d}{dh}F(x,y,z)+\mathrm{terms\ containing}\ h^{2}\end{array}$$

If we now diminish $$h$$ and increase $$M$$ without limit, their product $$Mh$$ remaining constant and equal to $$M'$$, the ultimate value of the potential of the pair of points will be

If f(x, y, z) satisfies Laplace’s equation, then $$V'$$, which is the difference of two functions, each of which separately satisfies the equation, must itself satisfy it.

If we begin with an infinite point of degree zero, for which

we shall get for a point of the first degree

$$\begin{array}{ll} V_{1} & =-M_{1}\frac{d}{dh_{1}}\frac{1}{r},\\ \\ & =M_{1}\frac{p_{1}}{r^{3}}=M_{1}\frac{\lambda_{1}}{r^{2}}\end{array}$$

A point of the first degree may be supposed to consist of two points of degree zero, having equal and opposite charges $$M_0$$ and $$-M_0$$, and placed at the extremities of the axis $$h$$. The length of the axis is then supposed to diminish and the magnitude of the charges to increase, so that their product $$M_{0}h$$ is always equal to $$M_1$$. The ultimate result of this process when the two points coincide is a point of the first degree, whose moment is $$M_1$$ and whose axis is $$h_1$$. A point of the first degree may therefore be called a Double point.

By placing two equal and opposite points of the first degree at the extremities of the second axis $$h_2$$, and making $$M_{1}h_{2}=M_{2}$$, we get by the same process a point of the second degree whose potential is