Origin of the Prismatic Colours. 405 



giving on emergence the number of waves expressed in (4) . 

 If V be the velocity in vacuum, t = \/V , and 



,_BC_AD 



so that 



T XV' 



Thus, as in (5), (6), (7), 



"-(*.- w^-^g • • ■ < 8 >< 



in agreement with (1). 



Although the process is less easy to follow, the construction 

 of a train of waves from an incident pulse is as definite in 

 the case of a prism as is that of a grating ; and its essential 

 features are presented to the eye in Scott Russel's phenomenon. 



The above treatment suffices for a general view, but it may 

 be instructive to give an analytical statement ; and this I am 

 the more inclined to do as affording an opportunity of calling- 

 attention to a rather neglected paper by Lord Kelvin entitled 

 " On the Waves produced by a Single Impulse in Water of 

 any Depth, or in a Dispersive Medium " *: When we know 

 the effect of an impulse, that of a uniform force applied for a 

 finite time can be deduced by integration. It may be con- 

 venient to recite the leading steps of Kelvin's investigation. 



Letf(k) denote the velocity of propagation corresponding 

 to wave-length (in the medium) 2irjk. The Fourier-Cauchy- 

 Poisson synthesis gives 



r 



Jo 



dk cos k[x-tf(k)] .... (9) 



for the effect at place and time (#, t) of an infinitely intense 

 disturbance at place and time (0, 0). When x — tf(k) is 

 yery large, the parts of the integral (9) which lie on the 

 two sides of a small range,- k — ol to k + ol, vanish by annulling 

 interference ; k being a value, or the value, of k, which makes 



3{A[*-*/(*)]}«o (io) 



or X=*t{f(K)+Kf'(K)} = TJt .... (11), 



* Proc. Roy. Soc. vol. xlii. p. 80 (1887). 



