Page:Scientific Memoirs, Vol. 1 (1837).djvu/466

(I)'

${\begin{aligned}k{\frac {dq}{dy}}&=-{\frac {dF}{dy}}+{\frac {dG}{dy}}+{\frac {dG_{1}}{dy}}+{\frac {dG_{2}}{dy}}\ldots \ldots +{\frac {dG_{\nu }}{dy}}+\mathrm {etc.} \\k{\frac {dq}{dz}}&=-{\frac {dF}{dz}}+{\frac {dG}{dz}}+{\frac {dG_{1}}{dz}}+{\frac {dG_{2}}{dz}}\ldots \ldots +{\frac {dG_{\nu }}{dz}}+\mathrm {etc.} \end{aligned}}$

which lead directly to the complete integral

(III)

$kq=C-F+G+G_{1}+G_{2}\ldots \ldots G_{\nu }+\mathrm {etc.} ;$

$C$ being an arbitrary constant.

In order to determine, by means of this equation, the density $q$, we must substitute for $F$, $G$, $G_{1}$, $G_{2}$, … $G_{\nu }$, &c. the integrals which they represent. If the rectangular co-ordinates are changed into polar co-ordinates by means of the known formulæ

${\begin{aligned}x&=r\sin \theta \cos \psi &y&=r\sin \theta \sin \phi &z&=r\cos \theta \\x'&=r'\sin \theta '\cos \psi '&y'&=r'\sin \theta '\sin \psi '&z'&=r'\cos \theta '\end{aligned}}$

the expression for $F$ takes the form (see the additions to the Connaissance des Temps for the year 1829, p. 356)

(IV)

${\begin{aligned}F&=\Sigma _{0}^{\infty }\left[{\frac {q}{r^{n+1}}}\iint \left(\int _{0}^{r}fq'r^{'n+2}\;dr'\right)P_{n}\sin \theta '\,d\theta '\,d\psi '\right]\\&+\Sigma _{0}^{\infty }\left[r^{n}\iint \left(\int _{r}^{\infty }{\frac {fq'}{r^{'n+1}}}\right)P_{n}\sin \theta '\,d\theta '\,d\psi '\right]\end{aligned}}$

The coefficient $P_{n}$ being given by the formula

$P_{n}={\frac {1.3.5..2n-1)}{1.2.3\dots \dots .n}}$

$\left\{p^{n}-{\frac {n(n-1)}{2(2n-1)}}p^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.3(2n-1)(2n-3)}}p^{n-4}+\mathrm {etc.} \right\}$

in which

$p=\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\psi -\psi '),$

and the limits of the integrals relative to $\theta '$ and $\psi '$ should be such that the value of $F$ may take in the whole space, except the small portions occupied by the material molecules.

In order to have the expression for $G$, let us in like manner put

$\xi =\rho \sin \omega \cos \phi ,\qquad \eta =\rho \sin \omega \sin \phi ,\qquad \zeta =\rho \cos \omega$

and represent by $\Pi _{n}$ the function $P_{n}$, when $r'$, $\theta '$, $\psi '$, are therein changed into $\rho$, $\omega$, $\phi$. Then, if we suppose the origin of the co-ordinates to be taken in the interior of the molecule, we shall have (see Connaissance des Temps for the year 1829, p. 357)

(V)

$G=\Sigma _{0}^{\infty }\left[{\frac {1}{r^{n+1}}}\iint \left(\int _{0}^{r}g{\bar {\omega }}\rho ^{n+2}\,d\rho \right)\Pi _{n}\sin \omega \,d\omega \,d\phi \right]$