Page:Scientific Memoirs, Vol. 2 (1841).djvu/208

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196

C. F. GAUSS ON THE GENERAL THEORY OF

filled, by a precise definition being given of what should be understood by a magnetic pole. We intend to apply this denomination to each point of the earth's surface where the horizontal intensity $=0$ : where therefore, speaking generally, the dip $=90$ ; but including the singular case, did it exist, where the total intensity $=0$ . If we were to give the name of magnetic poles to those places where the total intensity is a maximum (i. e. greater than anywhere in the surrounding vicinity), it must not be forgotten that this is something quite different from the above definition; that neither the situation nor the number of these last-named points have any necessary connexion with those of the points first spoken of; and that it tends to confusion when dissimilar things are called by the same name. If we look away from the actual condition of the earth and take the question in its generality, there may certainly exist more than two magnetic poles; but it does not appear to have been noticed that if, for example, two north poles exist, there must necessarily be between them yet a third point, which is likewise a magnetic pole, but is properly neither a north nor a south pole, or is both if that expression be preferred. A consideration of our system of lines will best serve to elucidate this subject. If the function $V$ have at a point of the earth's surface $P^{*}$ a maximum value $V^{*}$ , and all around smaller values, then a series of progressively decreasing values will correspond to a system of rings, each of which will inclose all the preceding ones, together with the point $P^{*}$ , and on each of these rings the direction of the horizontal magnetic force, or that of the north pole of the magnetic needle, will be inwards^[1].

This is the characteristic mark of a magnetic north^[2] pole.

It is clear that the rings may be made so small, or the corresponding values of the function $V$ may differ so little from $V^{*}$ , that any other point may be excluded.

We will designate by $S$ the space included by all the points on the surface of the earth at which the value of $V$ is greater

↑ These rings, themselves assumed as infinitely small, are not necessarily circular, but generally speaking oval, so that the normal direction of the magnetic needle in reference to them only coincides with the direction towards $P^{*}$ at four points of each ring. Great error may be involved, therefore, if without further precaution, the intersection of the prolongations of two compass directions at considerable distances is assumed to be $P^{*}$ .
↑ We conform here to the mode of speaking in common usage, according to which the point established by Captain James Ross is so designated, although properly speaking it is a south pole, when the earth itself is considered as a magnet.

[1] These rings, themselves assumed as infinitely small, are not necessarily circular, but generally speaking oval, so that the normal direction of the magnetic needle in reference to them only coincides with the direction towards $P^{*}$ at four points of each ring. Great error may be involved, therefore, if without further precaution, the intersection of the prolongations of two compass directions at considerable distances is assumed to be $P^{*}$ .

[2] We conform here to the mode of speaking in common usage, according to which the point established by Captain James Ross is so designated, although properly speaking it is a south pole, when the earth itself is considered as a magnet.

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