Origin of the Prismatic Colours. 403 



either in the structure of the medium or in the character of 

 the force, but the wave-length and velocity are variable 

 according to the direction considered. 



For the purposes of Scott Russel's phenomenon the localized 

 pressure is regarded as permanent ; but here it will be more 

 instructive if we suppose it applied for a finite time only. 

 Although the method is general, we may fix our ideas upon 

 deep water, subject to gravity (cohesion neglected), upon 

 which operates a pressure localized in a line and moving 

 transversely with velocity V. In the general two-dimensional 

 problem thus presented, the effect of the travelling pressure 

 is insignificant unless V is a possible wave-velocity ; but 

 where this condition is satisfied, a corresponding train of 

 waves is generated. In the case of deep water under gravity 

 the condition is always satisfied, for the wave-velocities vary 

 from zero to infinity. 



The limitation to a wave-train of velocity V is complete 

 only when the time of application of the pressure is infinitely 

 extended. Otherwise, besides the train of velocity V we have 

 to deal with other trains, of velocities differing so little from 

 V that during the time in question they remain sensibly in 

 step with the first. As is known *, the behaviour of such 

 aggregates is largely a matter of the group-velocity TJ, whose 

 value is given by 



u-^ ...... m, 



k being proportional to the reciprocal of the wave-length in 

 the medium. In the particular case of deep-water waves 



From this point of view it is easy to recognize that the 

 total length of the train of waves generated in time t' is 

 + (Y— (J)t'. If t be the periodic time of these waves, the 

 wave-length in the medium is Vt, and the number of waves 

 is therefore 



But for our present purpose of establishing an analogy with 

 prisms and their resolving-povver, what we are concerned 

 with is not the number of waves at any time in the dispersive 

 medium itself, but rather the number after emergence of the 

 train into a medium which is non-dispersive ; and here a 

 curious modification ensues. During the emergence the 



* See, for example, 'Nature/ xxv. p. 51 (1881) ; ' Scientific Papers/ 

 i. p. 540. 



2 F 2 



