Page:Scientific Memoirs, Vol. 2 (1841).djvu/217

${\begin{aligned}&Y=-{\frac {1}{\sin u}}\left({\frac {d\,P'}{d\,\lambda }}+{\frac {d\,P''}{d\,\lambda }}+{\frac {d\,P'''}{d\,\lambda }}+{},{\&}{\mbox{c.}}\right)\\&Z=2P'+3P''+4P'''+{},{\&}{\mbox{c.}}\end{aligned}}$

20.

If we combine, then, with these propositions, the known theorem, that every function of $\lambda$ and $u$, which, for all values of $\lambda$, from 0 to 360°, and of $u$, from to 180°, has a determinate finite value, may be developed into a series of the form

$P^{0}+P'+P''+P'''+{},{\&}{\mbox{c.}}$

the general member of which, $P^{n,}$ satisfies the above partial differential equation,—that such a developement is only possible in one determinate manner,—and that this series always converges,—we obtain the following remarkable propositions.

I. The knowledge of the value of $V$ at all points of the earth's surface is sufficient to enable us to deduce the general expression of $V$ for all external space, and thus to determine the forces $X$, $Y$, $Z$, not only on the surface of the earth, but also for all external space.

It is clearly only necessary for this purpose to develope ${\frac {V}{R}}$into a series according to the above-mentioned theorem.

In the sequel, therefore, unless it is expressly stated otherwise, the symbol $V$ is always to be taken as limited to the surface of the earth, or as that function of $\lambda$, and $u$ which follows from the general expression, when $r$ is made $=R$: thus

$V=R(P'+P''+P'''+{},{\&}{\mbox{c.}})$

II. The knowledge of the value of $X$ at all points of the earth's surface is sufficient to obtain all that has been referred to in Prop. I. In fact, according to Art. 15, the integral $\int _{0}^{u}X\,d\,u={\frac {V^{0}-V}{R}}$ signifying the value of $V$ at the north pole, and the developement of $\int _{0}^{u}X\,d\,u$ into a series of the form referred to must necessarily be identical with

$V^{0}-P'-P''-P''',{\&}{\mbox{c.}}$

III. In like manner, and under the considerations in Art. 16, it is clear that the knowledge of $Y$ on the whole earth, combined with the knowledge of $X$ at all points of a line run-