224-225] SUKFACES OF DISCONTINUITY. . 391 



It appears on examination that the undisturbed motion is stable or 

 unstable, according as 



&amp;lt; p coth kh+p coth kh 



&amp;gt; (pp coth kh coth kh }% 



where u is the velocity of the upper current relative to the lower, and c is 

 the wave-velocity when there are no currents (Art. 223 (ii)). When h and h 

 both exceed half the wave-length, this reduces practically to the former 

 result (10). 



225. These questions of stability are so important that it is 

 worth while to give the more direct method of treatment*. 



If &amp;lt;/&amp;gt; be the velocity-potential of a slightly disturbed stream 

 flowing with the general velocity U parallel to x, we may write 



&amp;lt;/&amp;gt; = - t/&quot;tf + &amp;lt;k ........................... (1), 



where fa is small. The pressure-formula is, to the first order, 



and the condition to be satisfied at a bounding surface y 77, where 



7? is small, is 



dri rjdrj_ dfa 

 dr U dx~~~fy ...... 



To apply this to the problem stated at the beginning of Art. 

 224, we assume, for the lower fluid, 



^ = aey+*&amp;lt;te+rf) ........................ (4); 



for the upper fluid 



&amp;lt;j )l = C e- k y + W x+ ^ ..................... (5); 



with, as the equation of the common surface, 



The continuity of the pressure at this surface requires, by (2), 

 p{i(&amp;lt;r + kU)C-ga}=p {i(&amp;lt;r + kU )C -ga} ...... (7); 



whilst the surface-condition (3) gives 



= -kC,\ 



.................... 



* Sir W. Thomson, &quot; Hydrokinetic Solutions and Observations,&quot; Phil. Mag., 

 Nov. 1871; Lord Kayleigh, &quot;On the Instability of Jets,&quot; Proc. Lond. Math. Soc. t 

 t. x., p. 4 (1878). 



