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The plan and objects of this series of papers have been so far explained already as to render it superfluous to enumerate them here. At the close of my former paper I have given the rectangular equation of the curve of verticity,—or that in any point of which a magnetic needle being placed, its line of direction would pass through the centre of the earth, and consequently be vertical to the horizon at the point where it cut the surface: but as the form and character of the curve could not be directly obtained from that equation, nor from any other into which it could be transformed; and as, moreover, the process by which they could be obtained required considerable preliminary investigations, I preferred to leave it in that state rather than give the partial and incomplete solution, which, in the midst of the deep domestic affliction that I was involved in, I must then have done. It is here, however, by pursuing a different course fully determined, as far, at least, as it is subservient to our physical problem: and I did not feel myself at liberty to insert in a paper on that subject any collateral inquiries, which, however interesting in a geometrical point of view, would be irrelevant to the immediate discussion professedly before me.

philosophical transactions of the royal society of london

IX. Geometrical investigations concerning the phenomena of terrestrial magnetism. Second series:— On the number of points at which a magnetic needle can take a position vertical to the Earth's surface