256 Wisconsin Academy of Sciences^ Arts, and Letters. 



Its vibrations are in the positive or the negative direction of rota- 

 tion in the plane of {x, y), according as ?z is positive or negative. 



The light is propagated in the positive or the negative direction 

 of the axis of z, according as n and q are of the same or of oppo- 

 site signs. 



In all media n varies when q varies, and — is always of the 



dq 



same sign with — . 



Hence, if for a given numerical value of n, the value of — 



is greater when ?i is positive than when n is negative, it follows 

 that for a given value of q, given both in magnitude and sign, the 

 positive value of n will be greater than the negative value. 



Now this is what is observed in a diamagnetic medium, acted 

 on by a magnetic force, y, in the direction of z. Of the two cir- 

 cularly-polarized rays of a given period, that is accelerated of 

 which the direction of rotation in the plane of (x, y) is positive. 

 Hence, of two circularly polarized rays, both left-handed, whose 

 wave-length within the medium is the same, that has the shortest 

 period whose- direction of rotation in the plane of {x, y) is positive, 

 that is, the ray which is propagated in the positive direction of z 

 from south to north. We have, therefore, to account for the fact 

 that when in the equations of the system q and r are given, two 

 values of n will satisfy the equations, one positive and the other 

 negative, the positive value being numerically greater than the 

 negative. 



We may obtain the equations of motion from a consideration of 

 the potential and kinetic energies of the medium. The potential 

 energy, Y, of the system, depends on its configuration, that Is, on 

 the relative position of its parts. In so far as it depends on the 

 disturbance due to circularly-polarized light, it must be a a func- 

 tion of r, the amplitude, and q, the coefficient of torsion, only. It 

 may be different for positive and negative values of q of equal 

 numerical value, and it probably is so in the case of media, which 

 of themselves rotate the plane of polarization. 



The kinetic energy, T, of the system, is a homogeneous function 

 of the second degree of the velocities of the system, the coefficients 

 of the different terms being functions of the coordinates. 



