356 Sir "William Thomson on Stationary 



or, with V 2 replaced by M/D 2 , 



where, as above, 



D3= a ° . . . (11) 



The elimination of b and D between these three equations 

 gives y as a function of/. It is clear that the change of level 

 of the bottom may be sufficiently gradual to obviate any of 

 the corrugational effect ; and when this is the case, the equa- 

 tion of the free surface will be found from y in terms of/; 

 / being a given function of the horizontal coordinate, a. 



If/ is everywhere small in comparison with a, D is approxi- 

 mately constant [much more approximately equal to J (a + b) ] , 

 and y is approximately in constant proportion to /. 



When the flow is so gentle that V is small in comparison 



]\/[2 



with sj9 D, --FT3 is a small proper fraction, and y is approxi- 

 mately equal to this fraction of/ 



Generally, in every case when V < V #D the upper surface 

 of the water rises, when the bottom falls, and the water falls, 

 when the bottom rises. 



On the other hand, when V> sj #D, the water surface 

 rises convex over every projection of the bottom, and falls 

 concave over hollows of the bottom ; and the rise and fall of 

 the water are each greater in amount than the rise and fall of 

 the bottom ; so that the water is deeper over elevations of 

 the bottom, and is shallower over depressions of the bottom. 



Returning now to the subject of standing waves (or cor- 

 rugations of the surface) of frictionless water flowing over a 

 horizontal bottom of a canal with vertical sides, I shall not at 

 present enter on the mathematical analysis by which the effect 

 of a given set of inequalities within a limited space AB of the 

 canal's length, in producing such corrugations in the water 

 after passing such inequalities, can be calculated, provided the 

 slopes of the inequalities and of the surface corrugations are 

 everywhere very small fractions of a radian. I hope before 

 long to communicate a paper to the Philosophical Magazine 

 on this subject for publication. I shall only just now make 

 the following remarks : — 



1. Any set of inequalities large or small must in general 



