Page:Scientific Memoirs, Vol. 1 (1837).djvu/467

${\begin{aligned}&+\Sigma _{0}^{\infty }\left[{\frac {1}{r^{n+1}}}\iint \left(\int _{0}^{r}g{\bar {\omega }}\rho ^{n}+2\,d\rho \right)\Pi _{n}\sin \omega \,d\omega \,d\phi \right]\\&+\Sigma _{0}^{\infty }\left[r^{n}\int _{'}\int _{'}\left(\int _{r}^{u}{\frac {g{\bar {\omega }}}{\rho ^{n-1}}}\right)\Pi _{n}\sin \omega \,d\omega \,d\phi \right].\end{aligned}}$

The double integral $\int ^{'}\int ^{'}$ is to be extended only to the points in respect to which the radius $u$ from the surface of the molecule is $<r$, and the integral $\int _{'}\int _{'}$ is to be extended to the points in respect to which $u>r$.

By means of a beautiful theorem which M. Poisson has demonstrated in the Memoir already quoted, and in the additions to the Connaissance des Temps for the year 1831, the functions given by the integrals

$\int _{0}^{r}g{\bar {\omega }}\rho ^{n}+2\,d\rho ,\qquad \int _{r}^{u}g{\bar {\omega }}\rho ^{n}+2\,d\rho ,\qquad \int _{r}^{u}{\frac {g{\bar {\omega }}}{\rho ^{n-1}}}\,d\rho$

may be represented by series of integer and rational functions of the spherical co-ordinates. Let $\Sigma H_{n}$, $\Sigma H_{n}^{'}$, $\Sigma H_{n}^{''}$, be these series; if the functions $H_{n}^{'}$, $H_{n}^{''}$ shall be found, so that they may be discontinued, and such that they are reduced to zero, the first for all the values of $u<r$, and the second for the values of $u<r$, we shall be able by means of the known theorems to reduce the expression for $G$ to the form

$G=\Sigma _{0}^{\infty }{\frac {4\pi }{2n+1}}\left({\frac {1}{r^{n+1}}}H_{n}+{\frac {1}{r^{n+1}}}H_{n}^{'}+r^{n}H_{r}^{''}\right).$

Such are the expressions for $F$ and $G$ which should be introduced into the equation (III). We might directly employ those which give the values of $G$, because they are always determinable when the position, figure, and density of the molecules are known; but the same thing cannot be done with the expression for $F$. This integral includes the quantity $q$, which is still unknown; and we should not be able to determine it by the condition that it would render the equation (III) identical without previously performing the integrations, an operation which would require the same function to be known. In order to avoid this difficulty, we are about to employ for the purpose of determining $q$ a differential equation corresponding with that marked (III), but in which the density $q$ is no longer included under the signs of integration.

The sum of these equations (I)', when they are differentiated in reference to $x$, $y$, $z$, respectively, gives

(VI.)

$k\left({\frac {d^{2}q}{dx^{2}}}+{\frac {d^{2}q}{dy^{2}}}+{\frac {d^{2}q}{dz^{2}}}\right)=4\pi fq.$