Page:Scientific Memoirs, Vol. 2 (1841).djvu/210

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If the junction take place at two different points, what has been demonstrated for one point would hold good for the two; and one may easily see that inside the space inclosing $$P^*$$ and $$P^{**}$$ an insular space would be formed, which would gradually contract itself as $$W$$ was diminished, and would necessarily at length resolve itself into a true south pole.

The case is similar when the junction takes place at three or more separate points; but if it take place at once on a whole line, then the horizontal force must disappear on all the points of that line.

It is evident that the assumption of two south poles would in like manner necessitate the existence of a third polar point, which would be neither a south pole nor a north pole, or rather would be both at once.

From what has been developed in the foregoing article, its application to many conceivable exceptions from the simplest type of our system of lines will be readily understood. The whole of the points to which a certain value of $$V$$ corresponds, may be a line consisting of several portions, of which each returns back into itself, but which are quite separate from each other; it may be a line crossing itself; lastly, it maybe a line having on both sides spat es where $$V$$ is greater than on the line, or where it is less.

We may assert that on the earth there are, on the great scale, no deviations of such a nature from the simplest type.