Page:Scientific Memoirs, Vol. 2 (1841).djvu/213

manner according; to $u$, ${\frac {d\,(V+RT)}{d\,u}}=0$; $V+RT$ having a value independent of $u$, or, what is the same thing, constant in all the points of a meridian,—it must hence also be absolutely constant, because all meridians converge and meet at the poles.

If we call the value of $V$ at the north pole $=V^{*}$, then $T={\frac {V^{*}-V}{R}}$;

and hence

$Y={\frac {d\,T}{\sin u.d\,\lambda }}$.

This result may also be expressed as follows:

$Y={\frac {1}{\sin u}}\int _{0}^{u}{\frac {d\,X}{d\,\lambda }}\centerdot d\,u.$

16.

This remarkable proposition, that, if the component of the horizontal magnetic force directed towards the north be given for the whole surface of the earth, then the component directed towards the west (or towards the east) follows of itself, is true, conversely, only with a certain modification. If $Y$ be expressed by a given function of $u$ and $\lambda$, and if $U$ represent the indeterminate integral $\int \sin u.Y\,d\,\lambda$ being assumed constant in the integration, then ${\frac {d\,(V+RU)}{d\,\lambda }}=0$, or $V+RU$ has a value independent of $\lambda$, and is, generally speaking, a function of $u$. Thus ${\frac {d\,(V+RU)}{R\,d\,u}}={\frac {d\,U}{d\,u}}-X$ is such a function; that is to say, the formula ${\frac {d\,U}{d\,u}}$ gives an imperfect expression for $X$, a part of it containing $u$ only remaining undetermined. This want would be supplied if, besides the expression for $Y$, we had also that for $X$, for some one given meridian, or to speak generally, for some line extending from the north to the south pole. We see therefore that, if we know the component of the horizontal magnetic force in the direction towards the west for the whole of the earth's surface, and the component in the direction towards the north for all points of some one line extending from the north pole to the south pole, the latter component, for the whole of the earth's surface, follows of itself.