Page:Scientific Memoirs, Vol. 2 (1841).djvu/216

From this equation, combined with the remark in the preceding article, we obtain the general form of $P^{(n)}$. If we represent by $P^{n,m}$ the following function of $u$,

${\Big (}\cos u^{n-m}-{\frac {(n-m)(n-m+1)}{2(2n-1)}}\cos u^{n-m-2}+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2\centerdot 4(2n-1)(2n-3)}}\cos u^{n-m-4}-{\&}{\mbox{c.,}}{\Big )}\sin u^{m}$

then $P^{(n)}$ has the form of an aggregate of $2n+1$ parts,

$P^{(n)}=g^{n,0}P^{n,0}+(g^{n,1}\cos \lambda +h^{n,1}\sin \lambda )P^{n,1}+(g^{n,2}\cos 2\lambda +h^{n,2}\sin 2\lambda )P^{n,2}+{\mbox{,}}{\&}{\mbox{c.}}+(g^{n,n}\cos n\lambda +h^{n,n}\sin n\lambda )P^{n,n},$

where $g^{n,0}$, $g^{n,1}$, $h^{n,1}$, $g^{n,2}$, &c. are determinate numerical co-efficients.

19.

If the magnetic force at the point $O$ be resolved into three forces perpendicular to each other, $X$, $Y$, and $Z$, of which $Z$ is directed towards the centre of the earth, and $X$ and $Y$ are tangential to a spherical surface concentric with the earth, passing through $O$, $X$ directed northwards in a plane passing through $O$ and the axis of the earth, and $Y$ directed westwards in a plane parallel to the equator of the earth, then

$X=-{\frac {d\,V}{r\,d\,u}},\quad Y=-{\frac {d\,V}{r\sin u\,d\,\lambda }},\quad Z=-{\frac {d\,V}{d\,r}};$

consequently,

${\begin{aligned}&X=-{\frac {R^{3}}{r^{3}}}\left({\frac {d\,P'}{d\,u}}+{\frac {R}{r}}\centerdot {\frac {d\,P''}{d\,u}}+{\frac {R^{2}}{r^{2}}}\centerdot {\frac {d\,P'''}{d\,u}},{\&}{\mbox{c.}}\right)\\&Y=-{\frac {R^{3}}{r^{3}\sin u}}\left({\frac {d\,P'}{d\,\lambda }}+{\frac {R}{r}}\centerdot {\frac {d\,P''}{d\,\lambda }}+{\frac {R^{2}}{r^{2}}}\centerdot {\frac {d\,P'''}{d\,\lambda }},{\&}{\mbox{c.}}\right)\\&Z={\frac {R^{3}}{r^{3}}}\left(2P'+{\frac {3RP''}{r}}+{\frac {4R^{2}P}{r^{2}}},{\&}{\mbox{c.}}\right)\end{aligned}}$

On the surface of the earth $X$ and $Y$ are the same horizontal components which we have designated above by those letters; $Z$ is the vertical component, which is positive when directed downwards.

The expressions for these forces on the surface of the earth are, then,

$X=-\left({\frac {d\,P'}{d\,u}}+{\frac {d\,P''}{d\,u}}+{\frac {d\,P'''}{d\,u}}+{},{\&}{\mbox{c.}}\right)$