Recent Progress in Theoretical Physics. 263 



These forces arise from the action of the remainder of the 

 medium on the element under consideration, and must in the case 

 ■of an isotropic medium be of the form indicated bj Cauchy, 



X= JLo ^. + Ai^ + etc. (12) 



Y=A,p^+A,p^ + etc., (13) 



dz^ dz^ 



If we now take the case of a circularly-polarized ray for which 



f = rcos {nt — qz), tj = r sin {nt—qz), (14) 



iwe find for the kinetic energy in unit of volume 



T = i prhv' ~ Cp^fn ■ (15) 



and for the potential energy in unit of volume 



F= r\A,<f - A,q' + etc.) = r'Q, (16) 



where ^ is a function of q^. 



The condition of free propagation of the ray given in equation 



(6), is 



dT^^dV 



dr dr 



which gives 



pn^ - 2Crin = Q, (18) 



whence the value of n may be found in terms of q. 



But in the case of a ray of given wave-period, acted on by mag- 



dq 

 netic force, what we want to determine is the value of -t~ when n 



is constant, in terms of — i, when y is constant. Differentiating (18) 

 dn 



{2pn - 2Crq') dn - | ^ + 4:0rqn I dq - 2Cqhidr = 0. (19 ) 



We thus find ±=- — ^^, ^. (20) 



dy pn — Cyq^ dn 



If X is the wave-length in air, and i the corresponding index of 

 refraction in the medium 



qX = ^Tii. nX = liiv. 



The change in the value of q, due to magnetic action is in every 



•case an exceedingly small fraction of its own value, so that we 



may write 



dq 



2 = ^0 + ^^' 



