Page:Scientific Memoirs, Vol. 2 (1841).djvu/214

17.

The foregoing investigations apply only to the horizontal portion of the earth's magnetic force. In order to embrace the vertical force also, we must consider the problem in all its generality; therefore $V$ must be regarded as a function of three variable magnitudes, expressing the position in space of an undetermined point $O$. We select for the purpose the distance $r$ from the centre of the earth, the angle $u$ which $r$ makes with the northern part of the earth's axis, and the angle $\lambda$, which a plane passing through $r$ and the axis of the earth makes with a first meridian, counted as positive towards the east.

Let the function $V$ be expanded into a series, decreasing according to the powers of $r$, and to which we give the following form:

$V={\frac {R^{2}P^{0}}{r}}+{\frac {R^{3}P'}{r^{2}}}+{\frac {R^{4}P''}{r^{3}}}+{\frac {R^{5}P'''}{r^{4}}}$, &c.

The co-efficients $P^{0}$, $P'$, $P''$, &c. are here functions of $u$ and $\lambda$; in order to see how they are connected with the distribution of the magnetic fluid in the earth, let $\mathrm {d} \,\mu$ be an element of the earth's magnetism, $\rho$ its distance from $O$, and let $r^{0}$, $u^{0}$, $\lambda ^{0}$, signify for $\mathrm {d} \,\mu$ the same as $r$, $u$, $\lambda$ for $O$. We have then $V=-\int {\frac {\mathrm {d} \,\mu }{\rho }}$ extended so as to include every $\mathrm {d} \,\mu$; further $\rho ={\sqrt {\;}}(r^{2}-2rr^{0})\cos u\cos u^{0}+\sin u\sin u^{0}\cos(\lambda -\lambda ^{0})+r^{0}r^{0}$, and if ${\frac {1}{\rho }}$ be developed in the series,

${\frac {1}{\rho }}={\frac {1}{r}}(T^{0}+T'{\frac {r^{0}}{r}}+T''.{\frac {r^{0}r^{0}}{r^{2}}}+{\&}{\mbox{c.}})$

then

${\begin{aligned}&R^{2}P^{0}=-\int T^{0}\,d\,\mu ,\quad R^{3}P'=-\int T'r^{0}d\,\mu ,\\&R^{4}P''=-\int T''r^{0}r^{0}\mathrm {d} \,\mu ,\;{\&}{\mbox{c.}}\end{aligned}}$

As $T^{0}=1$, and as according to the fundamental supposition with which we set out, the quantities of positive and of negative fluid are equal in every measureable particle in which they exist, and therefore are equal in the whole earth; that is to say, $\int \mathrm {d} \,\mu =0$, it follows that

$P^{0}=0$,

or the first number of our series for $V$ goes out.