22 Prof. Larmor, On the dynamical significance of 



And Lord Rayleigh has verified (Proc. Math. Soc. 1877; 'Theory of 

 Sound,' vol. i, Appendix) by a direct analysis of high generality that 

 the energy of the undulations is in fact conveyed with this velocity. 

 Thus if the medium could be such that the frequency diminished 

 with diminishing wave-length, the energy would travel backwards, 

 namely in a direction opposite to that in which its wave-form is 

 propagated. The question considered by Schuster and Lamb is 

 as to whether this is actually possible. Examples are given by 

 Prof. Lamb of infinitely extended trains in which it occurs. But 

 such a train requires to be fed with energy, so to speak, at both 

 ends. With a train of light advancing across a dispersive medium 

 from one side only, the case cannot occur, for the energy cannot 

 become negative *. 



The curve of dispersion is usually constructed with the fre- 

 quency v, equal to t _1 , as abscissa, and the index of refraction yu, 

 as ordinate. If c is the velocity of light in a vacuum 



Thus the group velocity U is cdv/d (/u,v). Hence 

 c d (uv) dix 



This must be positive ; thus dfjujdv can indeed be negative, but 

 the greatest allowable negative value at any point is — fi/v. So 

 far then as this present condition goes, the dispersion curve may 

 trend downwards, but to a limited extent, outside a band of 

 absorption, whereas upward trend is unrestricted. The condition 

 that a source of radiation must be emitting, not absorbing, energy 

 thus allows the dispersion curve to be an undulating line in a 

 region not containing absorption bands, but with a limit to the 

 steepness of the downward steps of the curve. In a region of 

 absorption this restriction is not imposed ; hence the trend will 

 usually be downward. Thus the character of the curve of -dis- 

 persion would be in the main that remarked by Kundt, though 

 the present consideration by itself does not require an upward 

 trend in transparent regions to be invariable. 



2. The general analytical verification of the rate of propaga- 

 tion of energy by Lord Rayleigh, above mentioned, prompted by a 

 more special discussion by Prof. 0. Reynolds, involves use of the 

 proposition that the mean energy, in stationary or progressive 



* This argument would be evaded if we could suppose that the wave-form 

 travels backward: but in the case of a train which is not endless, the wave-form is 

 of necessity propagated forwards, that is, in the direction in which the front of the 

 train advances. 



