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

$V'=-M'{\frac {d}{dh}}f(x,y,z).$

(9)

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

$V_{0}=M_{0}{\frac {1}{r}}$

(10)

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

${\begin{array}{ll}V_{2}&=-h_{2}{\frac {d}{dh_{2}}}V_{1},\\\\&=M_{2}{\frac {d^{2}}{dh_{1}dh_{2}}}{\frac {1}{r}},\\\\&=M_{2}{\frac {3\lambda _{1}\lambda _{2}-\mu _{12}}{r^{3}}}.\end{array}}$

(12)