Superposed Streams of Viscous Liquids. 495 



The complete solution will be given by 



where ty, yjr' are due to small disturbances, whose squares 

 are to be neglected. 



We assume i/r to be of the form 



On substitution for M* in (1), we obtain 



v (|? - / "T F( ' /) ~( a+ ikB + i * Cy X|? - * 2 ) Ffe) = °- (2) 



Writing 



@-^)F(.y) = /M 



we can put (2) into the form 



"(I? ~ ") /W ~ (a + '* B + ^M = °" • (3 > 



§3. Before proceeding further with the solution of the 

 problem. I wish to insert here the solution giving the form 

 of the free-surface of a stream of uniform depth flowing 

 over a corrugated bed, over which it is assumed that the 

 liquid can flow without experiencing any resistance. 



In equation (2) we have = 0, a = 0. 



It is easily shown that 



F(?/) = a x e k y + &!*?-** + a. 2 e x y + h 2 e~^, 

 where 



X» = ** + iifeB/v. 



We may take the bed to be 



y = -*+&**», 



and the free-surface 



y _ £,,«* 



Writing down the usual conditions 



V = - ttyfc. ^ = at y = - A + £ e*», 

 ^ = 0, /^ = 0, p yy = const, at ?/ = £*** z , 

 and eliminating the constants, we obtain the relation 



