242 COMPARISON OF THE ELECTRIC THEORY OP LIGHT 



case. Therefore, in finding the differential equation between I and p t 

 we may treat b and A' in (24) and / and G in (25) as constant, as well 

 as B and F. These equations may be written 



Differentiating, we get 



or 



Hence, if we write V for the wave-velocity (l/p), n for the index of 

 refraction, and X for the wave-length in vacuo, we have for the ratio 

 of the two parts into which we have divided the potential energy on 

 the elastic theory, 



bh* TT 2 Bh 2 _dlogV _ d\ogn 

 4 " I* ~dlogp~ ~dlog\' 



and for the ratio of the two parts into which we have divided the 

 kinetic energy on the electrical theory, 



7T 



fh* . TrFW cZlogV_ d log n 



____ _. (27) 



p 2 dlogp cHogX 



It is interesting to see that these ratios have the same value. This 

 value may be expressed in another form, which is suggestive of some 

 important relations. If we write U for what Lord Rayleigh has 

 called the velocity of a group of waves,* 



U_ 



V = " 



dlogl 



^281^1=5. (28) 



dlogp U 



It appears, therefore, that in the elastic theory that part of the 

 potential energy which depends on the deformation expressed 



*See his "Note on Progressive Waves," Proc. Lond. Math. Soc., vol. ix, No. 

 reprinted in his Theory of Sound, vol. ii, p. 297. 



