Page:Scientific Memoirs, Vol. 2 (1841).djvu/216

 204 From this equation, combined with the remark in the preceding article, we obtain the general form of $$P^{(n)}$$. If we represent by $$P^{n,m}$$ the following function of $$u$$, then $$P^{(n)}$$ has the form of an aggregate of $$2n + 1$$ parts, where $$g^{n,0}$$, $$g^{n,1}$$, $$h^{n,1}$$, $$g^{n,2}$$, &c. are determinate numerical co-efficients.

If the magnetic force at the point $$O$$ be resolved into three forces perpendicular to each other, $$X$$, $$Y$$, and $$Z$$, of which $$Z$$ is directed towards the centre of the earth, and $$X$$ and $$Y$$ are tangential to a spherical surface concentric with the earth, passing through $$O$$, $$X$$ directed northwards in a plane passing through $$O$$ and the axis of the earth, and $$Y$$ directed westwards in a plane parallel to the equator of the earth, then consequently, On the surface of the earth $$X$$ and $$Y$$ are the same horizontal components which we have designated above by those letters; $$Z$$ is the vertical component, which is positive when directed downwards.

The expressions for these forces on the surface of the earth are, then,