140 J.W. Gibbs — Comparison of the Electric Theory of 



length will also vary, and presumably to some extent the type 

 of vibration. But, by the principle just stated, if the ether or 

 the electricity could be constrained to vibrate in the original 

 type, the variations of I and p would be the same as in the 

 actual case. Therefore, in finding the differential equation be- 

 tween I and p, 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 



n* 

 47r 2 B^r + h P* = 47r °~ A '> 



I , TV'f i /-i 



ttF- + -4 = i G - 

 p* p* 



Differentiating, we get 



4tt 2 B^|- = - bd{ 2 f), 



TtF d- = ~ **fd(P~*)i 



or 



P~ 



P ,i w P 



4tt 2 B y d log L. = - bp 2 d logp\ 



tcY— # log _ = - -4 d logp 2 . 

 p> p> p 



Hence, if we write Y 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, 



M 2 ^_ ?r 2 BA 2 __ dlogV _ _ d log n .^ 



4 P d log p d log A.' 



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

 the kinetic energy on the electrical theory, 



^,/fc* __ nYFtf __ d log V . _ d log n .^ 



2? p 2 d log p> d log A. 



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

 This value may be expressed in another form, which is sug- 

 gestive of some important relations. If we write U for what 

 Lord Rayleigh has called the velocity of a group of waves,* 



5 = 1 



V 



rflogV 

 d log I ' 



d log V 



V-U 



d\ogl V 



d loo- V V-U 



(28) 



d\og2) U 



* See his Note on Progressive Waves, Proc. Lond. Math. Soe., vol. ix, No. 125, 

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



