229-230] REDUCTION OF THE INTEGRALS. 407 



so that, if we take the principal value of the integral in (ii), the surface-form 

 on the side of .^-positive is 



(xvi). 



Hence at a distance from the origin the deformation of the surface consists 

 of the simple-harmonic train of waves indicated by the first term, the wave 

 length ZTrhja being that corresponding to a velocity of propagation c relative 

 to still water. 



Since the function (ii) is symmetrical with respect to the origin, the 

 corresponding result for negative values of x is 



(xvii). 



The general solution of our indeterminate problem is completed by adding 

 to (xvi) and (xvii) terms of the form 



Ccosax/h 4- Dsin axfh ...... ... ............... (xviii). 



The practical solution including the effect of infinitely small dissipative 

 forces is obtained by so adjusting these terms as to make the deformation of 

 the surface insensible at a distance on the up-stream side. We thus get, 

 finally, for positive values of x, 



(xix), 

 and, for negative values of #, 



For a different method of reducing the definite integral in this problem we 

 must refer to the paper by Lord Kelvin cited below. 



230. The same method can be employed to investigate the 

 effect on a uniform stream of slight inequalities in the bed.* 



Thus, in the case of a simple-harmonic corrugation given by 



y = h-\- 7 cos kx (1), 



the origin being as usual in the undisturbed surface, we assume 

 (f)/c = % + (& cosh ky -f ft sinh ky) sin kx, 



(2) 

 y + (a sinh ky + ft cosh ky) cos kx { &quot; 



The condition that (1) should be a stream-line is 



7 = a sinh kh + ft cosh kh (3). 



* Sir W. Thomson, &quot; On Stationary Waves in Flowing Water,&quot; Phil. Mag., Oct. 

 Nov. and Dec. 1886, and Jan. 1887. 



