242 depend upon their relative position. Although his results are extremely interesting from a mathematical and theo retical point of view, we do not see that much practical advantage would attend the use of these equivalent poles ; and we are inclined to think that, except in the popular usage for distinguishing one end of a magnet from the other, and in the case of ideal magnets, the word pole had better be abandoned altogether. Distribu- The idea of representing the action of a linear magnet tion in a by a continuous distribution of free magnetism, positive in linear Qne j^jf an( j ne g a ti v e in the other, is very old. It appears in Bazin s work on the magnetic curves published in 1753; and Tobias Mayer, in his memoir above quoted, assumes that the density of the distribution is proportional to the distance from the middle of the bar. Four distinct methods have been used in attempting to determine the law of distribution. 1. The deflexions of a small needle in different positions near the magnet have been observed, and by means of these the constants in some formula assumed for the dis tribution have been calculated. This was the process adopted by Lambert and Hansteen, and, in some of his experiments, by Lamont. 1 2. Instead of measuring deflexion, we may count the oscillations of the needle, and proceed as before. This method was used by Coulomb, Becquerel, and Kupfer, but it led to no satisfactory results, partly owing to the disturb ances arising from induction and the force of translation upon the needle, partly owing to the difficulty of putting a satisfactory theoretical interpretation upon the results. Different 3. Some observers have measured the force required to methods detach a small armature of soft iron or steel from different e n " , parts of the bar, thinking thereby to obtain a direct measure of the free magnetism. It is not very easy to say what is measured by this process, but it is obvious, on a little consideration, that the effect is complex, depending greatly on the nature and extent of the surfaces in contact, and also upon the mutual induction between the magnet and the armature. Experiments of this kind have been made by Dub, Lamont, and others. 4. Another method frequently employed is to slide along the bar a small ring-shaped coil embracing it as closely as possible, and to measure the induction currents for a given displacement. The assumption usually made is that the integral electromotive force is proportional to the free magnetism on the portion of the bar passed over, or, what amounts to the same thing, to the difference between the magnetic moments per unit of length of the sections of the bar on which the coil rests at the beginning and end of the motion. This is, however, only an approximation to the truth, and the accuracy of this, approximation is very difficult to estimate in the practical case where the lateral dimensions of the bar are finite. The following investiga tion will show the nature of the difficulty. The integral electromotive force is - (N - N 2 ), where Nj and N 2 are the surface integrals of magnetic induction taken over the coil in its initial and final positions. Let us take first a linear solenoid SN (fig. 31) of length I, and magnetic moment m, and a coil of a single winding PQ, which moves so that its centre R is always in the line SN, and its plane always perpendicular to SN ; then the former integral extending all over PQ, the latter over the infinitely small section of the solenoid at R, a being the force due to the end distribution at N and S. We thus get i - cos0 ) (70), where 6 and are the angles PSX and PNX. This shows, in the first place (see equation (25) above), that if the coil PQ were to expand and contract as it moves, so as always to remain a section of the same tube of force, there would be no variation of N, and no electromotive force, which is as it should be. If we were at liberty to suppose PQ infinitely small, then, when R is between S and N, cos 6 - cose would be the sum of two unities, and, when R is outside, the difference. In such a case, so long as PQ moved on the magnet, there would be no electromotive force, but if we suddenly move it over the end, there would be an electromotive force See also Airy and Stuart, Phil. Mag., 1873. Fig. 31. Q - 4irm/7, which is proportional to the moment of the bar. When PQ is not infinitely small, there is a variable part of N, depending on the dimensions of PQ, which will give rise to an electromotive force, even when the coil is moved along a uniformly magnetized bar, where there is no free magnetism except at the ends. It is now easy to form a conception of what happens in the case of an ordinary complex solenoidal bar. We may suppose such a bar made up of a number of simple linear solenoids. A certain number of these, corresponding to the end parts of Poisson s distribution, will have the same length as the bar ; the others, corresponding to the lateral surface and volume parts of the distribution, will be of continuously diminishing lengths. If we were at liberty to suppose the lateral dimensions of the bar and the radius of the coil to be infinitely small, then, as the coil moves along the bar, we should have an electromotive force due to passage over the ends of the short solenoids, and, as it moves over the end, an electromotive force due to passage over the ends of the long solenoids. We might in this way by a sufficient number of observations determine the distribution of the free magnetism throughout the bar and at its ends ; and in this case no distinction would be necessary between the volume and the surface distribution in any section. If, however, the dimensions of the section of the bar, and con sequently of the coil, be finite, a correction would have to be applied, depending, not only on the dimensions of the bar and coil, but also on the magnetic distribution. All that we can then do is to assume a formula for Gauss s surface distribution and determine its con stants. We thus get Gauss s distribution, and a formula that will account for the electrical observations ; but we obtain no information as to the actual internal distribution of the magnetism in tlu bar. Lenz and Jacobi 2 appear to have been the first to apply the method of induction currents to the measurement of the magnetic distribution in bar magnets. They attempted no theoretical analysis of their results, although they assigned a law of distribution. Van Rees, 3 who questioned their conclusions, gave an imperfect theory, and made some careful researches of his own. Rothlauf 4 made further experiments, and entered more fully into the theory, though still with insufficient generality. The most recent experi ments of the kind we are aware of are those of Schaper, 5 who discusses the theory with complete generality, taking account of the ends of the bar. After what has been said, the reader will scarcely be Differei surprised to find that the different experimenters assigned laws as very different formulae for the distribution in bar magnets. Slgne Lambert deduces from his experiments a distribution whose density is A#, A being a constant, and x the distance from the ends of the bar. Brugmans, V. Swinden, and Lenz and Jacobi adopt the law Az 2 ; Hansteen, as we have seen, the law Ax r , where r-2 or 3. Biot deduced from Coulomb s experiments the law A(/i* -/* *) for the density of the free magnetism, which would give for the moment per unit of length of the bar the law a -&(ju.* + /*"*), see above, p. 231. Becquerel, 2 Pogg. Ann., Ixi., 1844. 3 Over de Verdeeling van het Magnetismus in Magneten, Amst., 1847.
4 Pogg. Ann., cxvi., 1862. 5 Wied. Ann., ix., 1880.