Page:Scientific Memoirs, Vol. 1 (1837).djvu/468

and it being observed that

${\frac {d^{2}F}{dx^{2}}}+{\frac {d^{2}F}{dy^{2}}}+{\frac {d^{2}F}{dz^{2}}}=-4\pi fq,\qquad \qquad {\frac {d^{2}G}{dx^{2}}}+{\frac {d^{2}G}{dy^{2}}}+{\frac {d^{2}G}{dz^{2}}}=0$

with respect to which see the third volume of the Bulletin de la Société Philomatique, p. 388.

If in this equation we change the differentials taken relatively to the rectangular co-ordinates into differentials taken relatively to the polar co-ordinates, we have

(1)

$k\left\{{\frac {d^{2}rq}{dr^{2}}}+{\frac {1}{r^{2}\sin \theta }}{\frac {d\left(\sin \theta {\frac {drq}{d\theta }}\right)}{d\theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {d^{2}rq}{d\psi ^{2}}}\right\}=4\pi frq.$

Let us suppose that $r\,q$ is developed in a series of integer and rational functions of the spherical co-ordinates, so that we may have

(2)

$rq=Q_{0}+Q_{1}+Q_{2}\ldots \ldots +Q_{i}+\mathrm {etc.} ;$

in which any one of the quantities $Q_{i}$ renders identical the equation

(3)

${\frac {d\left(\sin \theta {\frac {dQ_{i}}{d\theta }}\right)}{\sin \theta \,d\theta }}+{\frac {1}{\sin ^{2}\theta }}{\frac {d^{2}Q_{i}}{d\psi ^{2}}}+i(i+1)Q_{i}=0$

On this supposition the equation (1) will be satisfied by taking in general

(4)

$k\left\{{\frac {d^{2}Q_{i}}{dr^{2}}}-{\frac {i(i+1)}{r^{2}}}Q_{i}\right\}=4\pi fQ_{i}.$

In order to integrate this differential equation of the second order^[1] let us take

$Q_{i}={\frac {Q_{i}^{(1)}}{r}}-{\frac {1}{i}}{\frac {dQ_{i}^{(1)}}{dr}},$

and consequently

${\begin{aligned}{\frac {dQ_{i}}{dr}}&=-{\frac {Q_{i}^{(1)}}{r^{2}}}+{\frac {1}{r}}{\frac {dQ_{i}^{(1)}}{dr}}-{\frac {1}{i}}{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}\\{\frac {d^{2}Q_{i}}{dr^{2}}}&=2{\frac {Q_{i}^{(1)}}{r^{3}}}-{\frac {2}{r^{2}}}{\frac {dQ_{i}^{(1)}}{dr}}+{\frac {1}{r}}{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}-{\frac {1}{i}}{\frac {d^{3}Q_{i}^{(1)}}{dr^{3}}}\end{aligned}}$

1. The integration of this equation with the second member negative has also exercised the ingenuity of the two illustrious geometers Plana and Paoli. See the Memoirs of the Academy of Turin, vol. xxvi., and those of the Italian Society, vol. xx.