Green, in the 17th section of his Essay, has given an investigation of the distribution of magnetism in a cylinder of finite length acted on by a uniform external force parallel to its axis. Though some of the steps of this investigation are not very rigorous, it is probable that the result represents roughly the actual magnetization in this most important case. It certainly expresses very fairly the transition from the case of a cylinder for which κ is a large number to that in which it is very small, but it fails entirely in the case in which K is negative, as in diamagnetic substances.
Green finds that the linear density of free magnetism at a distance as from the middle of a cylinder whose radius is a and whose length is 2l, is
where p is a numerical quantity to be found from the equation
The following are a few of the corresponding values of p and κ.
| κ | p | κ | p |
| ∞ | 0 | 1.802 | 0.07 |
| 336.4 | 0.01 | 9.139 | 0.08 |
| 62.02 | 0.02 | 7.517 | 0.09 |
| 48.416 | 0.03 | 6.319 | 0.10 |
| 29.475 | 0.04 | 0.1427 | 1.00 |
| 20.185 | 0.05 | 0.0002 | 10.00 |
| 14.794 | 0.06 | 0.0000 | ∞ |
| negative | imaginary |
When the length of the cylinder is great compared with its radius, the whole quantity of free magnetism on either side of the middle of the cylinder is, as it ought to be,
Of this ½pM is on the flat end of the cylinder, and the distance of the centre of gravity of the whole quantity M from the end of the cylinder is a/p.
When κ is very small p is large, and nearly the whole free magnetism is on the ends of the cylinder. As κ increases p diminishes, and the free magnetism is spread over a greater distance
