Page:Scientific Memoirs, Vol. 2 (1841).djvu/215

We see further that $P'$ has the form

$R^{3}P'=\alpha \cos u+\beta \sin u\cos \lambda +\gamma \sin u\sin \lambda$,

where $\alpha =-\int \cos u^{0}r^{0}\,\mathrm {d} \,\mu$, $\beta =-\int \sin u^{0}\cos \lambda ^{0}r^{0}\,\mathrm {d} \,\mu$, $\gamma =-\int \sin u\sin \lambda ^{0}r^{0}\,\mathrm {d} \,\mu$. Therefore, according to the explanation laid down in page 13 of the Intensitas Vis Magneticæ, $-\alpha$, $-\beta$, $-\gamma$ are the moments of the earth's magnetism, in relation to three rectangular axes, of which the first is the axis of the earth, and the second and the third are the equatorial radii for longitudes 0 and 90°.

The general formulæ for all co-efficients of the series for ${\frac {1}{\rho }}$ may be assumed as known; it is merely necessary for our purpose to remark, that in relation to $u$, $\lambda$, the co-efficients are rational integral functions of $\cos u.\sin u\cos \lambda$, and $\sin u\sin \lambda$, and of $T''$ of the second order, $T'''$ of the third, &c. It is the same as to the co-efficients $P''$, $P'''$, &c.

The series for ${\frac {1}{\rho }}$, and for $V$, converge, so long as $r$ is not less than $R$, or rather, not less than the half diameter of a sphere, which includes all the magnetic particles of the earth.

18.

The function $V$ being composed of $-\int {\frac {\mathrm {d} \,\mu }{\rho }}$, satisfies the following partial differential equation:

$0={\frac {r\,d^{2}\,V}{d\,r^{2}}}+{\frac {d^{2}\,V}{d\,u^{2}}}+\cot u.{\frac {d\,V}{d\,u}}+{\frac {1}{\sin u^{2}}}\centerdot {\frac {d^{2}\,V}{d\,\lambda ^{2}}}$,

which is only transformation of the well-known equation

$0={\frac {d^{2}\,V}{d\,x^{2}}}+{\frac {d^{2}\,V}{d\,y^{2}}}+{\frac {d^{2}\,V}{d\,z^{2}}}$,

where $x$, $y$, $z$ signify the rectangular co-ordinates of $O$. If we substitute the value of $V$,

$V={\frac {R^{3}P'}{r^{2}}}+{\frac {R^{4}P''}{r^{3}}}+{\frac {R^{5}P'''}{r^{4}}}+$, &c.,

it is clear that for the several co-efficients, $P'$, $P''$, $P'''$, &c., there will likewise be partial differential equations, of which the general expression is

$0=n(n+1)P^{(n)}+{\frac {d^{2}\,P^{(n)}}{d\,u^{2}}}+\cot u{\frac {d\,P^{(n)}}{d\,u}}+{\frac {1}{\sin u^{2}}}\centerdot {\frac {d^{2}\,P^{(n)}}{d\,\lambda ^{2}}}$.