MAGNETISM 231 from N draw a serii S of lines to the points of division on B, and from S a similar series to the points of division on A. These lines will form a network of lozenges the loci of the vertices of which will be lines of force, corresponding to (25) or (27) according as we mind of the reader acquainted with the analysis employed in hydro kinetical problems the close analogy that subsists between the two methods. In fact, by proper arrangement, every problem in the one subject can be converted into a problem in the other. For details we refer the reader to Thomson, who was, so far as we know, ,he first to work out this matter fully ; in the present connexion lie should consult more particularly g 573 sq. of the Reprint. Infin- itely small magnet, pass from point to point along one set of lozenge diagonals or along the other. Fig. 22 will give the reader an idea of the general appearance of the two sets of lines. He may compare the ideal with tlic actual cases by referring to figs. 4 and 5, p. 222. In the case of an infinitely small magnet, the equipotential lines are of course given by the polar equation r 2 = c 2 cosfl, c being a variable parameter. It is easily shown that the lines of force, which are , necessarily orthogonal to these, have for their equation r = c sin 2 ^. 1 If <f> be the angle between r and the tangent of the line of force, we have tan(p--=rde/dr=^ tanO ; hence the following construction for the direction of the line of force at P due to a small magnet at : let K be the point of trisection of OP nearest 0, and let KT, per- pa idicular to OP, cut the axis of the magnet in T ; then TP is the tangent to the line of force at P. This construction in a slightly diiferent form was given by Hansteen 2 and by Gauss 3 ; the latter ad Is that the resultant force at P is given by M. PT/OT. OP 3 where M is the magnetic moment of the magnet, a proposition which the readi-r will easily verify. These propositions are of considerable use in rough magnetic calculations. As this is an important case we give a diagram of the equipotential lines and lines of force in fig. 23. We may, if we choose, consider a filament of matter magnetized longitudinally at every point, but so that the strength I (=J, say) is variable. Such a filament is called a complex solenoid. It may clearly be supposed made up of a bundle of simple solenoids whose eii ls are not all coincident with the ends of the filament. If ds be an element of such a filament, the potential is given by Tint is, its action may be represented by two particles of magnetism Jj and J 3 at its two ends, and by a continuous distribu tion of free magnetism along its length whose density is -dijds. This is of course merely a particular case of Poisson s distribution. When a body is solenoidally magnetized, the magnetic force H both external and internal depends solely on the surface distri bution, i.e., merely on the ends of the solenoids of which the body is composed. We may therefore suppose the two ends of any solenoid joined by a solenoid of equal strength lying in the surface of the body. Proceeding thus, we may in an infinite number of ways construct a surface layer of tangentially magnetized matter which will represent the magnetic action of a solenoidally magnetized body. Thomson has shown by means of a highly interesting piece of analysis how to find the components of this tangential magnetization. See Reprint of Papers on Electricity and Magnetism, p. 401. The magnetic theorems just stated will suggest at once to the i Hansteen, Magnetismus der Erde, p. 208 (1819).
- Maynrtitmux der Erde, p. 209.
3 Retaliate d. Mag. Vereins, 1837 and 1840. Fig. 23. Uniformly Magnetized Bodies constitute in practice the mo^t im- Potential portant case of solenoidal magnets. In the first place it is obvious of uni- that the whole magnetic moment of such a body is simply its volume formly multiplied by the intensity of magnetization, and that the axis of magnet- the whole is parallel to the axis of each of its infinitely small parts, ized The method usually applied to calculate the potential in this case bodies, maybe presented in two ways. The potential is calculated accord ing to Poisson s method in pf this case merely from a surface distribution of vary ing density I cos 6. We may replace this by a layer of uniform density p and vary ing normal thickness. Let the thickness at any point measured parallel to the magnetic axis be t ; then the normal thickness is cosfl ; hence pt cos 6 Icos0, and pt = I ; i.e., t is constant. We may there fore suppose the magnet replaced by itself (iig. 24) with a uniform volume distribution p of positive magnetism, and itself dis placed through a distance t in a direction opposite to that of magnetization with a uniform volume distribution - p ; or, which comes to the same thing, the potential of the magnet at P is p(U - U ), where U is the potential at Pof a uniform volume distri bution of density + 1 throughout the magnet, and U the potential of the same at a point P displaced through a distance t in the direction of magnetization. If I, m, n be the direction cosines of the magnetic axis, this gives at once (29); - i - dy AX + BY + CZ where X, Y,Z are the components of the resultant force due to volume distribution p= + 1 throughout the body, and A,B,C the components of the magnetization. The same result may also be arrived at thus. The part of the potential due to the element dv is Idv cos 0/r 2 , but this is the com ponent parallel to the direction of I of the resultant force at P of a
clume distribution whose density in dv is I ; hence, since the