Page:Scientific Memoirs, Vol. 1 (1837).djvu/473

from this last equation, that is to say, that which is given by the formula

(2)′

$r(q-q_{0})=Q_{0}+Q_{1}+Q_{2}\ldots \ldots +Q_{i}+\mathrm {\&c.}$

This being premised let us return to the formula (5). As the integrations indicated in the second member of this equation may, according to what we have stated at the commencement of this paragraph, be extended from $r'=0$ to $r'=\infty$, $\theta '=0$ to $\theta '=\pi$, and $\psi '=0$ to $\psi '=2\pi$, and as all these limits are independent of each other, observing that we have in general

${\begin{aligned}&\int _{0}^{\pi }\int _{0}^{2\pi }P_{n}T_{i}^{'}\sin \theta '\,d\theta '\,d\psi '=0\\&\int _{0}^{\pi }\int ^{2\pi }P_{n}V_{i}^{'}\sin \theta '\,d\theta '\,d\psi '=0\end{aligned}}$

and in particular when $i=n$;

${\begin{aligned}&\int _{0}^{\pi }\int _{0}^{2\pi }P_{n}T_{n}^{'}\sin \theta '\,d\theta '\,d\psi '={\frac {4\pi }{2n+1}}T_{n};\\&\int _{0}^{\pi }\int _{0}^{2\pi }P_{n}V_{n}^{'}\sin \theta '\,d\theta '\,d\psi '={\frac {4\pi }{2n+1}}V_{n}.\end{aligned}}$

we shall find

${\begin{aligned}F&=\Sigma _{0}^{\infty }{\frac {4\pi }{2n+1}}{\frac {f}{r^{n+1}}}\left\{T_{n}\int _{0}^{r}\Omega _{n}(r^{\prime })r^{\prime n+1}\,dr^{\prime }\right.\\&\left.\qquad +V_{n}\int _{0}^{r}\Omega _{n}^{\prime }(r^{\prime })r^{\prime n+1}\,dr^{\prime }\right\}\\&+\Sigma _{0}^{\infty }{\frac {4\pi }{2n+1}}fr^{n}\left\{T_{n}\int _{0}^{\infty }{\frac {\Omega _{n}(r^{\prime })}{r^{\prime n}}}dr^{\prime }\right.\\&\qquad +\left.V_{n}\int _{r}^{\infty }{\frac {\Omega _{n}^{\prime }(r^{\prime })}{r^{\prime n}}}dr^{\prime }\right\}\end{aligned}}$

Without actually making the substitutions of the expressions previously given for $G$, and latterly for $F$, in the equation (III)' for the purpose of comparing the functions of the spherical coordinates of the same degree which are to render it identical, we see that, as $G$, $G_{1}$, $G_{2}$, &c., contain none of these functions, all the $T_{n}$ and $V_{n}$ must be null, with the exception of $T_{0}$ and $V_{0}$, which answer to the value $n=0$, and represent two arbitrary constants.

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