Page:Scientific Memoirs, Vol. 1 (1837).djvu/469

By making the substitutions in the equation (4) we shall be able to exhibit the result in the following form:

$k\left[{\frac {1}{r}}\left\{{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}-{\frac {(i-1)i}{r^{2}}}Q_{i}^{(1)}\right\}-{\frac {1}{i}}\left\{{\frac {d^{3}Q_{i}^{(1)}}{dr^{3}}}-{\frac {(i-1)i^{(1)}}{r^{2}}}{\frac {dQ_{i}^{(1)}}{dr}}+{\frac {2i(i-1)}{r^{2}}}Q_{i}^{(1)}\right\}\right]={\frac {4\pi f}{r}}Q_{i}^{(1)}-{\frac {4\pi f}{i}}{\frac {dQ_{i}^{(1)}}{dr}}.$

The foregoing equation is satisfied by taking

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

This equation is of the same form with that proposed (4), except only that $i$ is replaced by $i-1$. If therefore again, in the latter, we put

$Q_{i}^{(1)}={\frac {1}{r}}Q_{i}^{(2)}-{\frac {1}{i-1}}{\frac {dQ_{i}^{(2)}}{dr}}$

we shall deduce from it another in terms of $Q_{i}^{(2)}$, in which $i-1$ will be replaced by $i-2$; and by continuing these substitutions we shall finally obtain the equation

$k{\frac {d^{2}Q_{i}^{(i)}}{dr^{2}}}=4\pi fQ_{i}^{(i)};$

which is integrable by the known methods, and gives

$Q_{i}^{(i)}=T_{i}e^{r{\sqrt {\frac {4\pi f}{k}}}}+V_{i}e^{-r{\sqrt {\frac {4\pi f}{k}}}}$

where $T_{i}$ and $V_{i}$ may be considered as two arbitrary functions of $\theta$ and $\psi$ of the order $i$ which satisfy the equation (3).

By adopting this value $Q_{i}^{(i)}$, and by afterwards taking

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

$Q_{i}^{(i-2)}={\frac {1}{r}}\quad Q_{i}^{(i-1)}\;-\;{\frac {1}{2}}{\frac {dQ_{i}^{(i-1)}}{dr}}$

$Q_{i}^{(i-3)}={\frac {1}{r}}\quad Q_{i}^{(i-2)}\;-\;{\frac {1}{3}}{\frac {dQ_{i}^{(i-2)}}{dr}}$

$\dots \dots \dots$

$\dots \dots \dots$

$\dots \dots \dots$