ON THE ACTION OF MAGNETISM ON LIGHT. 34T 



equation of motion of the form K-jj ^^ ^''' — ''^l^ere f and \p each stand 



for one of the symbols E, r), 'C, and s stands for one of the symbols a;, y, z, — 

 and if we trace backwards the operation of integration by parts by which. 

 it was derived from the characteristic function, we obtain the following- 

 types under the sign of volume-integration which may exist in the direct 

 variation of that function : 



£^ d^ d0 d^xP d^ d^ . d^SxP 

 dsdt ds ' dt ds^ ' ds dsdi* ds^dt 



These expressions may combine into complete variations of terms of any 

 of the types 



d^d4 ^ ^ . d^ 

 dsdt ds' dt ds^ ' ds'^dt 



Now the term in the energy which comes from the linking of the optical 

 with the magnetic motion should be of the first degree as regards the 

 velocities of each of them ; and it may involve linear and angular dis- 

 placements, but not their differential coefiBcients, i.e., it should involve 

 only second differential coefficients with respect to space. The first of 

 these types is thus the only one available, and the term in the energy 



must therefore be a scalar constructed from the combination of — with, 



dt 



|a,,0,and(/,p,;.)or(|-^^, ..., ...), 



if we exclude the scalar 



dE dri ,d^ 



dx dy dz ' 



which would introduce compression. The term under investigation may 

 therefore have the form either 



•'dOdt ^ dMt dtidt 



or 



dE df dt] dg , d^ dh 

 dd dt Mlt dH dt' 



excluding, for the reason already given, forms such as 



Of these the first combines together the angular distortion of the medium, 

 the velocity representing the motion in the magnetic field, and the rate 

 of change of the velocity of the medium in the direction of that motion; 

 while the second combines the s^piv of the medium with the velocity in 

 the magnetic field. It would be difficult to assign a physical basis to the 

 former on either a dynamical or an electric theory ; and thus we are 

 confined on our premisses to Maxwell's form as giving correctly the 

 localisation of the magneto-optic part of the energy. This dynamical 

 conclusion if granted will restrict the purely formal results of § 6, in the 



