TEANSACTIONS OF SECTION A. 559 



of small gyrostats, having the undisturbed positions of their axes in the commoo 

 direction of the magnetic force and the propagation of tlie beam, and all ■vibrating 

 in the same sense. When in consequence of the vibrating motion each gyrostat 

 has its axis of rotation displaced from this direction, it reacts on the surrounding 

 medium with transverse force at right angles to the plane through the axis of 

 rotation and the direction of motion. 



By compounding this stress with the elastic forces of displacement of the ether, 

 differential equations of motion are obtained which are of precisely the form neces- 

 sary to account for the diS'erenoe in rate of propagation of the two circularly 

 polarised rays constituting the plane polarised ray. 



It is obviously suggested by the gyrostatic investigation that it ought to be 

 possible to explain the magneto-optic rotation on the electromagnetic theory of 

 light as a consequence of the existence of the small magnets which are supposed 

 imbedded in the medium with their axes in the direction of propagation of the ray, 

 and therefore producing the magnetisation which the medium has in that 

 direction. 



In consequence of the motions of the ether, the direction of the chains of 

 magnetised molecules which are supposed to exist along the direction of magnetisa- 

 tion (here taken as axis of s) in the undisturbed state of the medium is continually 

 undergoing change at every point, and thus the direction of the axial magnetic 

 force along each chain also undergoes alteration. It is obvious that if the dis- 

 placements be everywhere small, the actual magnitude of this force will sustain 

 only a very small percentage of alteration, but that each small change of direction 

 will produce a component magnetic force in each of the two directions at right 

 angles to the axis. The calling into existence of these components will produce 

 corresponding electromotive forces tending to increase the displacements. 



The electromotive force in the direction of y is given by 



Q = _dG_d± 

 (It dy 



where dG\dt stands for the total time rate of change of G, the component of vector 

 potential in the direction of y. Also since II, the component along z, does not 

 perceptibly vary along .r, if the direction of propagation be as taken here along z, 

 — 8Ct/9s denotes magnetic induction through unit of area in the plane of yz. Hence 

 any part of the total time-rate of variation of —dG\dz will denote the space-rate of 

 variation in the direction of z of an electromotive force parallel to y, provided the 

 time and space differentiations of the part are commutative. 



Now if the displacement of the ether j^articles from the undisturbed positions 

 be take7i as parallel and proportional to the electric displacement, and be the 

 component of magnetisation of the substance in the direction of s due to the exis- 

 tence of the molecular magnets, then considering the electric displacement / in the 

 direction of .r, we see that the component magnetic force in the direction of x is 

 eCdfldz, and thus the magnetic induction through unit of area in the plane of yz is 

 fieCdfldz, where e is a coefficient of proportionality. The time-rate of variation of 

 this is 



-4,1 



But we have by the equations of electric currents 



di 47T\dy dzj 4tt dz 



since there is no conduction current. 



Further, by the relation of magnetic force to vestcr potential, i3 = (oF/9s)//x, 

 and therefore the last equation becomes 



^t' 4;r/xS^ 



