THE MAGNETIC FIELD. 19 we try to consider it so, the want of reality in our conception of lines of force becomes at once apparent. This will be seen from the following consideration. If, as we assume, a mechanical force can only be exerted by lines actually passing through the magnet pole, it will be evident that in case the pole be a mathematical point, only one line can pass through it and exert mechanical force on it. This force would therefore be quite independent of the density of lines around the pole. If the pole, although of the same strength, had finite dimensions, more lines would actually pass through it, and more mechanical force would be exerted. Experiment, however, shows that this is not the case, and that within reasonable limits the mechanical force is independent of the extent of the pole, and only depends on its free magnetism. From this we conclude that a strictly geometrical representation of the density of lines in a magnetic field, in the same manner as we might represent the density of trees in a forest, would be incorrect. We cannot pretend to solve the problem of finding a geometrical representation for our conception of the intensity of the magnetic field, and we must be content to use the term in its conventional sense, without having any clear idea of how it could be represented by a mechanical model. Yet this is no reason why we should abandon such an extremely convenient method of representing magnetic action at a distance. Nobody has as yet succeeded in explaining the action of gravitation, or has been able to represent it by a mechanical model. Nevertheless we find no difficulty in using the conventional terms of acceleration, mass, and weight of bodies in our calculations. We know that the weight of a body equals the product of its mass and the acceleration due to gravity. If we put strength of pole for mass and intensity of field for acceleration due to gravity, we find the analogue to weigh in the mechanical force with which a free magnet pole is acted on when placed in a magnetic field. From what has been said above, it will be evident that we must define magnetic field of unit intensity as that in which a free magnet pole of unit strength is acted on with unit force. To define a magnet pole of unit strength we must have recourse to the well-known expression for the mechanical attraction or repulsion existing between two poles placed at a certain distance apart. The law has been established experimentally by Coulomb,' with the aid of his torsion balance, and verified by Gauss,2 who used for the purpose a large fixed magnet, and a smaller suspended magnetic needle. It is as follows. M and m denote the strength of the two poles, and if they are placed at a distance, r, from each other, the mechanical force (attraction or repulsion according to whether the poles are of dissimilar or similar sign) acting between M m them is If both poles are equal and of the strength m, we have It and if their distance be unity, the force acting between them will equal the square of the free magnetism of one pole. The force will be unity if the free magnetism is unity. We find, therefore, the definition for unit pole to be a pole of such strength that when placed at unit distance from an equal pole, the two will act upon each other with unit force. It remains to define unit force and unit distance. This might be done on any convenient basis of the measurements of mass, length, and 1 Wüllner, "Experimentalphysik,” iv., § 5. FUNDAMENTAL UNITS. 21 time. In electrical calculations it is customary to use for this purpose The Gram as the unit of mass. The Centimeter as the unit of length. The Second as the unit of time. On these units is based what is known as the Absolute System of Electro-Magnetic Measurements. Taking these units as the basis for our calculations, we can find all other units of measurement, because they are all connected in some way with the fundamental units of mass, length, and time. We find thus that the unit of velocity is one centimeter per second, that of acceleration is an increase of velocity of one centimeter per second, and since mechanical forces are measured by the product of mass and acceleration, we define the unit of mechanical force, the Dyne, as that force which applied to a mass of one gram, during one second, will give it a velocity of (or accelerate its velocity by) one centimeter per second. The mechanical energy represented by the force of one dyne acting through a distance of one centimeter is the unit of energy, and is called the Erg. Having accepted these fundamental and derived units, we can now proceed to establish units for the lines of force, and for the intensity of the magnetic field. We call a unit line of force one of such strength that if a unit pole be placed on it, it shall be urged along it with the force of one dyne. A unit magnetic field would be one in which a unit pole would be acted on with the force of one dyne. If we find experimentally that an equal force is exerted in all points of a certain portion of the field (as is the case with the magnetic field of the Earth within certain limits), we say that this particular portion of the field is of uniform magnetic intensity, and we consider all the lines of force to be straight, parallel, and equidistant. A uniform magnetic field of unit intensity is therefore one in which every square centimeter of transverse section is traversed at right angles by one unit line. We can now determine the number of unit lines which emanate from a free unit pole. Before doing so, a few words of explanation regarding this conception of a free magnet pole are necessary. It has been shown above that magnets are produced by the adjusting of their molecules into continuous chains; and that, therefore, equal quantities of magnetic matter of opposite signs are produced at the poles of the magnet. Experiment shows that it is physically impossible to produce a magnet with one pole only, and that therefore no such thing as a free magnetic pole can be found in nature. But we can get an approximation to the free pole by making the magnet very long in comparison to the strength of its poles. In this way the magnetic influence of each pole will be sensibly felt through a distance considerably smaller than the length of the magnet, and when investigating the magnetic properties of the space immediately surrounding one pole we can neglect the disturbing influence of the other pole. In this case the lines of force emanating from the pole under consideration will be straight radii, shooting out from the pole all around into space. Let, in Fig. 4, P be the pole, and S a sphere described around it as centre, then this sphere will be pierced by the lines of force, in points which are all equidistant from the pole. Let r be that distance, and M the strength of the pole, we find the mechanical attraction exercised upon a unit pole of opposite sign placed at any point on the surface of the sphere, by the expression M If, now, a second sphere be described around P, with a ABSOLUTE MEASUREMENTS. 23 radius larger than r by only an infinitesimal amount, we shall have a spherical shell of infinitely small thickness, within which the intensity of the field is uniform. Into whatever point between the two surfaces of the shell we may place our unit pole, we find that it is attracted with the same force towards P, and from this we conclude that the density of lines all over the spherical surface must be uniform. Since in a uniform field the force exerted upon unit pole in the direction of the lines is equal to their Fig. 4. M p2 density (or number of lines per square centimeter of transverse section), we conclude that through each square centimeter of surface on the sphere, there pass unit lines. Now the total surface of a sphere of radius r, is 4 π r2, and consequently the total number of lines emanating from the pole of the strength M is If the pole P, instead of having the strength M, were a |