Superposed Streams of Viscous Liquids. 501 



(a) Cp>cy. 



Now when k is very small 



pg±pO +(Cp _ CV) ] 2 _ s 



= ic / k(p + pi){2(Cp-C'p')-( P - P , )(C + C')}/(C + V/) 



= igk(p+p')\C-C')l{C + G'). 



Thus the numerator can be negative when C<C, but the 

 denominator can never be negative. Hence the motion is 

 unstable for disturbances of great wave-length when C<C. 

 Remembering that Cp>C'p', and that vpC=v / p / G\ these 

 conditions are equivalent to T/p' <vp, and v'>v. These two 

 inequalities are not inconsistent with p>p', and hence under 

 these circumstances the interface will be unstable for waves 

 propagated in one direction of length greater than some 

 limit. 



(b) Cp<G'p>. 



As before, the denominator is always positive. The 

 numerator corresponding to one mode will always be nega- 

 tive when k is small, and in consequence the interface will 

 be unstable. Since vpG = v'p f Q f , we have shown that, if 

 v' <v, the motion is unstable for waves of great wave-length. 



Thus putting the two cases together we have shown that, 



if v'p'<vp, »/>v, 



or if v' < v, 



the motion will be unstable for great wave-lengths. Taking 

 into account the fact that p'<p, these two cases are both 

 included in the inequality v'p' <vp. 



There still remains the question of the validity of the 

 assumption that ikO is small in comparison with u + ikB, 

 or a . Now from the expression for « it is quite evident 

 that this is satisfied, except when k is very large, without 

 the necessity of assuming that C is small, if the c. G. s. system 

 of units be used. 



It is to be noticed that (a + ikB) and ikKj are not of the 

 same dimensions, and therefore the assumption that ikC is 

 small compared with (a + i&B) may not lead to a solution valid 

 when y is great. But this does not affect the form of the 

 solution when y is small, which is all we require to know for 

 the present purpose. 



Clare College, Cambridge. 



