264 Lord Bayleigh on Waves. 



(A) 



_ d(f) _ dty 

 dx dy 



_ d(f> _ dyjr 

 dy dx 



Hence, if the bottom of the canal be taken for axis of x, we may- 

 take for u and v, since they satisfy Laplace's equation, 



« = co S (4)/(.,)=/- ^- +T -^_ /IT _ & c.,| (fi) 



-v=^( 1/ £) A , )=u r- I i^ 3r+ ..., J 



where f(x) is the slowly variable value of u at the bottom 

 when y = 0, and accents indicate differentiation with respect 

 to x. The corresponding expression for ty is 



*~^-rfa/ /,+ 1.2.3.4.5 /"- • <P> 



This equation applies to the upper boundary, if we understand 

 by ty the there constant value of the stream-function, and gives 

 us a relation between the ordinate of the boundary and the 

 function/. 



If p be the pressure at the upper surface, we have 



where C is some constant. We will write for brevity, 



t*+^-«-tyy; (D) 



and the object of the investigation is to examine how far it is 

 possible to make -cr constant by varying the form of y as a 

 function of x. Since u 2 + v 2 = (1 +y /2 )u 2 , our equation becomes 



or, on substituting for u its value, 



JJ 1.2 y 1.2.3.4' 7 V 1+y 2 



Between this equation and (C), /may be eliminated by succes- 

 sive approximation ; and we obtain as the relation between y 

 and m, 



In this investigation y is regarded as a function of x, which 



