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

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204
C. F. GAUSS ON THE GENERAL THEORY OF

From this equation, combined with the remark in the preceding article, we obtain the general form of . If we represent by the following function of ,


then has the form of an aggregate of parts,


where , , , , &c. are determinate numerical co-efficients.

19.

If the magnetic force at the point be resolved into three forces perpendicular to each other, , , and , of which is directed towards the centre of the earth, and and are tangential to a spherical surface concentric with the earth, passing through , directed northwards in a plane passing through and the axis of the earth, and directed westwards in a plane parallel to the equator of the earth, then


consequently,


On the surface of the earth and are the same horizontal components which we have designated above by those letters; is the vertical component, which is positive when directed downwards.

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