Origin of the Prismatic Colours. 407 



agreeing with the general value given before, since here 

 XJ , = iy? If, as would be more convenient in order to find 

 the length of the train after emergence into a non-dispersive 

 medium, we regard x as given, we find that t ranges from 

 2xiVto2x/V + t'. 



I have taken the particular case first, as the reasoning is 

 rather simpler when we have, as in (13), an explicit expression 

 in terms of x and t*. In general /c cannot be eliminated 

 between (11) and (12), and we must proceed rather differently. 

 The question is when will 



*[*-*/(*)] c 17 ) 



with 



*=*{/(*)+*/'(*)] (18) 



be stationary with respect to f, x—Yt r being substituted for x 

 and*-*' for fin (17), (18)? Now 



(vi + ,>)-(v-/ W>+ (v£ + g)(.,-.^F> 



of which the second term on the right vanishes by (18). The 

 variation of (17) vanishes whenV=/(&). Accordingly 



*[«-w-(i-o/(*)] 



with 



x-Vt'=(t-t'){f(k) + kf'(k)} 

 is stationary with respect to t', if V =/(&), and then assumes 

 the form 



k[x-Yt]. 



Here t' must lie between and t'. Thus if t be constant, 

 x has a range 



Yt'-t'{f(k) + kf'(k)}={Y-TJ)t f . 

 And if x be given,, t has a range 



W _ *'(U-V) 



/(*)+*/.'(*) u ' ' ■ 



These are the limits over which the waves of velocity V 

 extend. And (19) shows that the number of waves which 

 pass a fixed point, either within the dispersive medium or on 

 emergence from it, has the expression 



n_ u-v 



where r is the periodic time, in agreement with (4). 



* For an admirable discussion of the general problem of deep-water 

 waves arising from a localized disturbance, see Lamb, Proc. Lond. Math. 

 Soc. vol. ii. p. 371 (1904). 



