ON LINES OF INDUCTION IN A MAGNETIC FIELD. 
319 
Section III.— Mathematical Appendix. 
Brief History and General Solution for Elliptic Cylinders and Confocal Elliptic 
Shells. 
(a) The foundation of the modern mathematical theory ot magnetic induction was 
laid by Poisson, between 1821 and 1838. He was the first to work out in detail the 
solution for a solid or hollow sphere of paramagnetic material placed in a field of 
uniform intensity. # Subsequently, he extended his investigations to the case of an 
ellipsoid placed in a uniform field.! Green, at a later date, gave an approximate 
solution for a cylinder of finite length placed with its axis along a uniform field. In 
1848, J. Neumann attacked the problem of an ellipsoid of revolution placed in any 
given field.j In 1854, Ivirchhoff succeeded in solving the same problem for a 
circular cylinder of infinite length. In 1881, A. G. Greenhill considered the case 
of a hollow ellipsoid. § 
Lord Kelvin was the first to publish, in 1872,[| diagrams of lines of induction for 
spheres of paramagnetic and diamagnetic material placed in a uniform field. On 
account of their frequent reproduction (they figure in almost every text-book on the 
subject), these diagrams are now very well known. In Maxwell’s great treatise 
are to be found a number of two-dimensional diagrams. These include the following 
cases:—(1) two circular cylinders rigidly magnetised transversely, and placed with 
their magnetic axes at right angles to each other (vol. 2, fig. 14); (2) a circular 
cylinder permanently magnetised in a transverse direction, and placed in a uniform 
field, so that the direction of magnetisation of the cylinder is coincident with that of 
the field; (3) a cylinder of diamagnetic material in a uniform field (vol. 2, fig. 15); 
(4) a permanently magnetised cylinder in a uniform field whose direction is at 
right angles to the direction of magnetisation of the cylinder (vol. 2, fig. 1G) ; 
(5) a uniform field disturbed by a current in an infinitely long straight cylindrical 
conductor normal to the direction of the field (vol. 2, fig. 17). 
In 1882 , StefanIT published a long paper dealing with the induced magnetisation 
of an infinitely long hollow circular cylinder. He obtains a solution by assuming the 
magnetic potential V to be of the form (Ar + B/r) cos r/>, where r is the distance of 
the point considered from the axis of the cylinder, and <£ the angle between r and 
the direction of the impressed field ; the constants A* and B assuming different 
values for the regions within the cylinder, in its substance, and outside the cylinder 
respectively. The function A = Ax + Bx/r 3 is an integral of the equation 
of the com 
* ‘ Maxwell,’ vol. 2, p. 59. t Ibid., p. 66. 
I ‘ Crclle,’ Bel. 37 (1848). § ‘ Journal de Physique ’ (1881). 
|| ‘ Reprint of Papers on Electrostatics and Magnetism,’ pp. 493-495. 
^1 ‘ Wien. Ber.,’ 85, Part 2, p. 613. 
dpV 
d.A 
03V 
+ = 0, and it is the real part of the function A z + B/ 
