18 Proceedings of the Roijal Irish Academy. 



If we further suppose that, as a function of t, v is proportional to e"'', where 

 n is a real or complex constant this becomes 



{n + kV) {dHlcly" - JcH) - hvdr Ujdif = 0. (7) 



Lord Eayleigh devotes special attention to various cases in which the 

 stream is composed of several separate layers in each of which the rotation in 

 the steady motion is constant, but a different constant for different layers. 

 He regards Z7 as continuous, some cases in which Z7is supposed discontinuous 

 having been discussed by him in a pre%dous paper " On the Instability of Jets."* 



If, in any layer, the rotation Z is constant, d^ U/dy^ = 0, and, wherever 

 n + kU is not equal to zero, (7) reduces to 



S-^'-O. (8) 



The solution of this is 



V = Ad'y + Be-^y, (9) 



where A and B are constants, real or complex. For each layer of constant 

 Z, a fresh solution with different constants is to be taken, the partial solutions 

 being fitted together by means of the proper conditions at the surfaces of 

 transition. One of these conditions is 



Av = 0. (10) 



Another is obtained by integrating (7) across the surface of transition, and is 



{n^kU).^'^-kv./^'^ = 0. ■ ai) 



dy dy 



This last equation, to be satisfied at the fixed plane which is the separating 

 surface in the steady motion, expresses the condition that there shall be no 

 slipping at a surface of transition. At first sight it might appear that this 



condition requires 



^{U+^t) = " (12) 



at the fixed plane in question. WhoX is required, however, is that (12) should 

 be satisfied at the disturbed surface ; and it may be shown that this reduces 

 to (11). This may be seen as follows : — Let the surface of separation be 



F = y - hcoB {nt + kx) = 0, 

 and suppose on the positive side 



U+^o = U+ 2Zy + [Ae^y - Be'^y) cos {nt + kx\ 



V = [Ae^y + Be-^y) sin {nt + kx), 

 and on the negative 



U+u = U+ 2Zy + {A'd'y - B'e-^v) cos {nt + kx), 



V = {A'd'y + B'e-^y) sin {nt + kx). 



* P. L. M. S. X., p. 4, 1878 ; Scientific Papers, t. i., p. 361. 



