Constitution of Natural Radiation. 583 



source with a velocity equal to the difference of these two 

 velocities, and Lord Rayleiglr's determination of the length of 

 disturbance emitted in a given time ensues, — subject, however, 

 to the reservation quoted above from Lord Kelvin as regards 

 the extreme head of the train. The account of the process, 

 which is indicated by Lord Rayleigh, seems in its essentials 

 to be fully verified. 



Yet the quotation above made from Sir George Stokes, in 

 which the imagery is optical instead of hydrodynamical, 

 appears to show a different aspect of the picture which we 

 are bound to follow out ; though Lord Rayleigh has guarded 

 himself against it by his reservation, " so long at least as we 

 are content to take for granted the character of the dispersive 

 medium — the relation of velocity to wave-length — without 

 inquiring further as to its constitution/'' The postulate thus 

 indicated is that the partial differential equation of propa- 

 gation is to hold true without limitation. This implies that 

 the dispersive medium must be homogeneous in space ; if it 

 had minute alternating structure, then this differential 

 equation could not of course be applied without modification 

 to waves of length comparable with the dimensions of: that 

 structure, — a circumstance on which Cauchy reared his 

 original attempt at an explanation of optical dispersion. But 

 it requires also that the medium should, so to speak, be 

 homogeneous in time. An optical dispersive medium is made 

 up of elements which have periods of free vibrations of their 

 own, that are more or less durable ; the differential equation 

 will not hold for disturbances whose scale of duration is so 

 small as to be of the same order as the time of the natural 

 subsidence of free disturbance among the elements of the 

 medium. In connexion (originally) with the dynamical 

 theory of viscosity in gases, Maxwell introduced the term 

 time of relaxation to express the time, roughly assignable, 

 that it would take for a local derangement of the molecules 

 of the medium to smooth itself out. In optics it is the time 

 needed for the free irregular vibrations of an element of the 

 medium, produced by a local shock or other disturbance, to 

 die out by dissipation into surrounding elements. The theory 

 of regular dispersion of a disturbance into wave- trains caused 

 by refraction, re-stated above for the hydrodynamic case of 

 waves on water, cannot be applied in the optical case unless 

 the scale of duration of the disturbance is long compared 

 with the time of optical relaxation of the dispersive medium. 

 In Sir George Stokes's illustration, taken at a venture, the 

 time of relaxation would be ten thousand times the period of 

 a light-wave ; if so, regular refraction and dispersion would 



