496 Mr. W. J. Harrison on the Stability of 



between ft and /3 in the form 



[P cosech kh — Q cosech \h~\/3 



= [P coth kh - Q ooth\h+gk 2 /v]/3, 

 where P = Jcv(2k 2 + t&B/j/) 2 , 



Q = U 4 \v, 



This leads to a finite expression for j3 for all velocities of 

 the stream, and also gives the difference of phase between 

 the corrugation and the wave-profile. When the wave- 

 length is small compared with the depth the solution differs 

 widely from that in the case of a non- viscous liquid*. 

 When the stream velocity is great the amplitudes in the 

 viscous and the non-viscous cases are equal. 



§1. There are two ways of rigorously solving the equation 

 (3). The first is that which was given by Lord Kelvin in 

 his paper on " Rectilineal Motion of Viscous Fluid between 

 two Parallel Planes" t- But in order to adapt this method 

 of solution to our problem it is necessary to employ very 

 complicated integration of Fourier's type. 



The second rigorous solution can be obtained in terms of 

 the Bessel's functions Ji, Ii ; this problem thus affords a 

 second example of the use of these functions in physical 

 analysis. The double integration involved in the solution of 

 equation (2) from that of (3) can be expressed, first of all, 

 as a triple integral and then reduced to a single integral by 

 the aid of Dr. J. W. Nicholson's results in his paper " On the 

 Relation of Airy's Integral to the Bessel's Functions" J. 



But again the analysis involved in the use of these 

 functions is too complicated, and we are forced to consider 

 an approximate solution §. 



§5. Returning to equation (2) we shall solve by successive 

 approximation on the supposition that ikC is small compared 

 with a-H'£B. To satisfy this assumption it is not necessary 

 in general that C should be small. 



* Cf. Lamb, Hydrodynamics, 3rd edition, p. 389. 



t Phil. Mag. xxiv. p. 192 (1887). 



% Phil. Mag. (6) xviii. pp. 6-17 (1909). 



§ Since writing this my attention has been drawn to two papers by 

 Prof. W. M'F. Orr in the ' Proceedings ' of the Royal Irish Academy, 

 1907, in which he uses these functions in the similar but less complicated 

 problems of the stability of motion of a single liquid flowing between 

 parallel planes. 



