the Stability of the Flow of Fluids. 69 



and this is equal to zero in virtue of the conditions at the 

 limits. Tn like manner the coefficient of /u is zero, as appears 

 on successive integrations by parts. The coefficient of p is 



-K(S p+ (©'+"' ♦•*■}*■ 



so that p = 0. 



Again, multiply (25) by /3, (26) by a, and subtract. On 

 integration as before the coefficient of q is 



/{©MS)**" +«*■}<* 



and that of fx is 



Hence q has the same sign as \i* y that is to say, q is positive. 

 That n in e int is a pure positive imaginary is no more than 

 might have been inferred from general principles, seeing that 

 the problem is one of the small motions about equilibrium of 

 a system devoid of potential energy. 



Since (24) is an equation with constant coefficients, the 

 normal functions in this case are readily expressed. Writing 

 it in the form 



{»-'•-;}{£-'•}■=»■ • • m 



we see that the four types of solution are 



Jcx g—kx Jk'x g—ik'x 



where 



-*«=*«+ m//»; . . . (28) 



or, if we take advantage of what has just been proved, 



k'* = q/n-k*, (29) 



where q and \l are positive. It will be seen that the odd and 

 even parts of the solution may be treated separately. Thus, 

 for the first, 



u = A sinh kx + B sin Jc'cc, .... (30) 



and the conditions to be satisfied at x= ±a give 



= A sinh ka + B sin Ua "1 ( ^\ 



= kA cosh ka + MB cos k'aj ; ' ' * ^ 0l) 



