Page:Scientific Memoirs, Vol. 2 (1841).djvu/211

Local deviations, indeed, may well be supposed to exist. Magnetic masses near the surface, though producing no sensible effect at any considerable distance, may obscure and even obliterate the regular progress of the terrestrial magnetic force in their immediate vicinity. In the simplest case the system of lines in such a district might take the form represented in Figure 2.

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14.

After this geometrical representation of the relations of the horizontal magnetic force, we proceed to develope the mode of submitting them to calculation. On the surface of the earth $V$ becomes a simple function of two variable magnitudes, for which we will take the geographical longitude reckoned eastward from an arbitrary first meridian,—and the distance from the north pole of the earth; we will designate the first of these, or the longitude, by $\lambda$, and the second, or the complement of the geographical latitude, by $u$. Considering the earth as a spheroid of revolution, of which the greater semi-axis $=R$, and the lesser semi-axis $=(1-\epsilon )R$, an element of the meridian is

$={\frac {(1-\epsilon )^{2}R.d\,u}{(1-(2\epsilon -\epsilon ^{2})\cos u^{2})}}{\frac {3}{2}}$;

and an element of the parallel is

$={\frac {R\sin u.d\,\lambda }{\sqrt {(1-(2\epsilon -\epsilon ^{2})\cos u^{2})}}}$