ON THE ACTION OF MAGNETISM ON LIGHT. 337 



We may in fact develope a complete and compact account of the 

 matter, as follows. The equations for the displacements in a circular 

 transverse vibration, propagated along the axis of z, are 



u= A cos (nt — ez), w^Asin (nt — ez) • 



for a given value of z these equations represent a circular vibration in the 

 plane of xy, and this is propagated in spiral fashion as a wave. We may 

 very conveniently combine the two equations into one by use of the vector 

 &=H + tv to represent the displacement, thus obtaining the form 



As this vibration is propagated without change, the equation of propaga- 

 tion must be linear in 3-, therefore of the form 



The terms in P involve higher differential coefficients, and are necessary 

 in order that the two values of e corresponding to a given value of n 

 may not be equal except as to sign, in other words in order that right- 

 handed and left-handed waves of the same period may be propagated at 

 different speeds. To ensure this result, P must contain terms of odd 

 order in the differential coefficients ; if there were only terms of even 

 order, it would still lead to an equation for the square of e, and so would 

 represent ordinary dispersion without the rotational property. 



If we confine our attention to terms of the first and third orders we 

 can tabulate possible rotational terms as follows 



1 



d33 (P^ d^ d^^ d^^ d^ 



Now in the case of the first three types, change of sign of z does not affect 

 the phenomenon ; thus the rotation is in the same direction whether the 

 wave travels forward or backward ; it is of the magnetic kind. In the 

 case of the last three types, change of sign of z produces the same effect 

 as change of sign of the rotatory coefficient ; the rotation is of the kind 

 exhibited by quartz and sugar and other active chemical compounds. 



On an ultimate dynamical theory, if •& denote displacement in a medium 



of density p^p will represent force per unit volume ; and the principle 



of dimensions shows that (Cj/p, ».'2/p> '-'a/P) are respectively of dimensions 

 [L^T-i], [T], [T-i], in length and time. Thus the coefficient k^ will pro- 

 duce rotation owing to some influence of a distribution of angular momen- 

 tum pervading the medium ; while the coefficients k^ and k^ would produce 

 selective rotation owing to the influence of the free periods of the fine- 

 grained structure of the imbedded atoms of matter. The latter kind of 

 rotation is to be expected to a sensible amount only in the rare cases in 

 which selective absorption of the light is prominent ; consequently we are 

 guided, as a first approximation, to ascribe magnetic rotation to a co- 

 efficient of the type Vj. 



The last three types of term will be appropriate to represent the 

 rotation of naturally active media. The dimensions of ^:^|p, K^/p, k-^/p, 



' J. Larmor, Proc. Lond. Math, Soc., xxi. 1890. 

 1893. z 



