190 Prof. Challis on the Hydro dynamical Theory of the 



21. I proceed now to account by the hydrodynamical theory 

 for the experimental facts on which the Gaussian argument for 

 the law of the inverse square in magnetism rests. In one set 

 of experiments the small magnet was placed so that the prolon- 

 gation of the axis of the large magnet passed through its centre 

 and cut its axis at right angles. Under these circumstances the 

 ordinate q = 0, so that the transverse velocity vanishes, and the 

 expression for the longitudinal velocity becomes 



kfjiu 3 / 1 1 \ 



which, if the ratio of / top be small, is very nearly 



p 3 

 In another set of experiments the small magnet was placed 

 with its axis pointing to the centre of the large one transversely 

 to the axis; in which case, since j» = 0, the transverse velocity 

 again vanishes, and, supposing the ratio of / to q to be small, 

 the approximate expression for the longitudinal velocity becomes 



_ kfiuH 



Hence in both cases the directive force varies inversely as the 

 cube of the distance from the centre of the large magnet, and at 

 equal distances is double in the former case to what it is in the 

 other. The two principal results of the experiments having 

 been thus accounted for, the hydrodynamical theory has effected, 

 at least to a first approximation, all that may strictly be de- 

 manded from it. In order, however, to exhibit its applicability 

 more fully, I shall now employ it to show why Gauss's empirical 

 theory succeeds in representing the same facts. 



22. It has been inferred from the hydrodynamical theory that 

 the action of the large magnet on the small one is simply directive. 

 Hence, assuming that each magnet has near its ends a positive 

 pole and a negative pole, and that like poles repel each other 

 and unlike poles mutually attract, it will readily be seen that, 

 according to the arrangements of the two experiments described 

 in art. 21, the actions on the poles of the small magnet are nearly 

 equal, and nearly in the same direction, and that the action on 

 one is attractive and that on the other repulsive. These forces 

 are, therefore, proper for acting as a kind of couple, and giving 

 direction to the axis of the needle. Also in this mode of viewing 

 magnetic action, if, as is empirically assumed, the force varies 

 inversely as some power of the distance from the pole, the law 

 of the inverse square is alone applicable, because experiment and 

 the hydrodynamical theory concur in indicating that the direc- 



