Page:Scientific Memoirs, Vol. 2 (1841).djvu/205

perly speaking, are only different modes of expressing the same thing) may be tested, at least approximately, by a reference to observation.

Let $P^{0},\,P',\,P''\ldots P^{0}$ be a polygon on the surface of the earth, the sides of which are the shortest lines that can be drawn between their respective extremities, and are therefore portions of great circles, the earth being here considered simply as a sphere. Let $\omega ^{0}$, $\omega '$, $\omega ''$, &c. be the intensities of the horizontal magnetic force at the points $P^{0}$, $P'$, $P''$, &c.; further, let $\delta ^{0}$, $\delta '$, $\delta '',$ &c. be the declinations reckoned in the usual manner, west of north as positive, east of north as negative; lastly, let $(01)$ be the azimuth of the line $P^{0}P'$ at $P^{0}$, reckoned in the customary manner, from the south by the west; in like manner $(10)$ the azimuth of the same line taken backwards at $P'$, and so on.

Let it be observed that $t$ alters continuously in each of the sides of the polygon, but suddenly at the corners, where therefore ithas two different values; for example, at $P$, $t$ has the value $(10)+\delta '$, in consideration that $P'$ is the end of the line $P^{0}P'$; and the value of $18^{rc}+(12)+\delta '$, in regard that it is the beginning of $P'P''$.

We may consider the approximate value of the integral $\int \omega \cos t.d\,s$, extended through $P^{0}P'$, to be

${\frac {1}{2}}(\omega ^{0}\cos t^{0}+\omega '\cos t').P^{0}P'$,

where $t^{0}$ and $t'$ signify the values of $t$ at $P^{0}$ as the beginning, and at $P'$ as the end of $P^{0}P'$. This approximation is all that can be obtained, because we have the values of $\omega$ and t</math> only at the extremities $P^{0}P'$, and is deserving of confidence in proportion to the shortness of the line. The given expression is, in our notation,

$={\frac {1}{2}}(\omega '\cos((10))+\delta ')-\omega ^{0}\cos((01))+\delta ^{0}]).P^{0}P'$.

In like manner, the approximate value of the integral, extended through $P'P''$, is

${\frac {1}{2}}(\omega '\cos((21)+\delta '')-\omega '\cos(12)+\delta ']).P'P''$,

and so on through the whole polygon.

Therefore, for a triangle our proposition gives the approximatively correct equation

${\begin{aligned}&\omega ^{0}(P^{0}P'\cos((01)+\delta ^{0})-P^{0}P''\cos((02)+\delta ^{0}])\\+&\omega '(P'P''\cos((12)+\delta ^{'})-P^{0}P'\cos((10)+\delta '])\\+&\omega ''(P^{0}P''\cos((20)+\delta '')-P'P''\cos((21)+\delta ''])=0.\end{aligned}}$