Page:Scientific Memoirs, Vol. 2 (1841).djvu/204

Let $V$ alter by infinitely small but equal steps. A system of surfaces will be produced, dividing space into infinitely thin strata, and the inverse ratio of the thickness of the strata to the intensity of the magnetic force will then hold good not only for different points in one and the same stratum, but also for different strata.

8.

We will now take into consideration the values of $V$ on the surface of the earth.

At a point $P$ of the earth's surface let $\psi$ be the intensity; $PM$ the direction of the whole magnetic force; $\omega$ the intensity, and $PN$ the direction of the force projected on the horizontal plane, or $PN$ the direction of the magnetic meridian, meaning thereby the direction indicated by the north pole of the magnetic needle; $i$ the angle between $PM$ and $PN$, or the dip; $\theta$, $t$, the angles formed by the elementary portion $d\,s$ of a line on the surface of the earth and the directions $PM$, $PN$. Lastly, $V$ and $V+d\,V$ correspond to the two extremities of $d\,s$.

We have consequently

$\cos \theta .=\cos i\cos t,\omega =\psi \cos i.$

And the equation in Art. V. becomes

$d\,V=\omega \cos t.d\,s$

If two points on the earth's surface $P^{0}$ and $P'$, at which $V$ has the value of $V^{0}$ and $V'$, are connected by a line traced on the surface of the earth of which $d\,s$ is an indeterminate element, then

$\int \omega \cos t.d\,s=V'-V^{0}$,

if the integration be extended through the whole line; and it is plain that three corollaries hold good similar to those in Art. VI., namely,

I. That the integral $\int \omega \cos t.d\,s$ keeps the same value by whatever path you proceed on the surface of the earth from $P^{0}$ to $P'$.

II. The integral $\int \omega \cos t.d\,s$ throughout the whole length of a closed line on the surface of the earth is always $=0$.

III. In such a closed line, unless throughout its course $t=90^{\circ }$, a part of the values of $t$ must necessarily be acute and a part obtuse.

9.

Propositions I. and II. of the foregoing article (which, pro-