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



This equation leads to the solution 



«o=±i[{(C/)-cy)H¥<p 2 -p' 2 )}H(c P -cy)]/2( /3 + /) ') 



=/3(say), 



like signs being taken together. 



There are two modes propagated in opposite directions 

 with different velocities relatively to the stream velocity at 

 the interface. If Cp = Op\ we obtain the usual result. 



For the purposes of the next approximation we have to 

 substitute 



V = (« /vO* + 5;Ky/4« , 

 n 2 — pot — 5ikCp/2(ot v)*y 

 ra 3 ' » p'ao + JLik&p'fifaify, 



n 3 = gPp/uQ, 

 n B ' = gPp'l^. 



In these approximate values the quantities R, S, T 7 W 

 are not sufficiently important to contribute any term, and 

 therefore might have been omitted. But this would have 

 given the solution the appearance of being correct to the 

 first power of ikC/(u + ikJ$) only, instead of to the second, 

 as we require it to be. 



To a second approximation a = /3-f 7, where it is quite 

 easily shown that 



7 = -ft 1 n W(P-/0W*[s^±(Cp-(y^)]* 



' - (p + p')KpSv' + P Wv) 



4 9 k( P +p')+(Cp-c/p')(c+c')±(C'+c)Sh 

 ± 9 k(p*-p>*)+(Cp-c/p'y±(c P -c'p')& ' 



where S = (Cp - Cy) 2 + tyKf-p'*) • 



With the exception of the first choice of signs, like signs 

 are to be taken together. 



This result can be verified by putting C = C / = 0. It then 

 agrees with the former result * for two superposed fluids at 

 rest, provided we take the negative sign in the first choice. 



The sign of the real part depends upon that of 



W(p +p') + (ty- CV)(C + CQ ± (C + OQS* 



- 4^*(^-^) + (Op-cy)«±(Cp-cy)s* * 



* Proc. Lond. Math. Soc, ser. 2, vol. vi. p. 399 (1908). 



