Waves in Flowing Water. 521 



(14) we see that Cf xh = (17) . 



and hence by the first of equations (16) we see that 



J , 1 — ztccosmx + K" ' 



*/ — 7T/OT 



Now when # is infinitely little short of unity the factor of 

 dx in the first member of (18) is zero for all values of x dif- 

 fering finitely from zero or 2i7r/m, (i being an integer); and 

 it is infinitely great when x = or ^iir/m. Hence we infer 

 from (17) and (18) that a vertical longitudinal section of the 

 bottom presents a regular row of symmetrical elevations and 

 depressions above and below its mean level; the elevations being 

 confined to very small spaces on the two sides of each of the 

 points # = and x=2i7r/m, and the profile-area of each eleva- 

 tion being A. The depths of the depressions below the average 

 level in the intermediate spaces between the elevations, are of 

 course extremely small because of the exceeding shortness of 

 the spaces over which are the elevations. For our complete 

 analytical solution, not only must A be infinitely small, but 

 the steepness of the slope up to the summit of A must every- 

 where be an infinitely small fraction of a radian ; and of 

 course therefore the infinitesimal lowering of the bottom be- 

 tween the ridges, which the adoption of a mean bottom-level 

 for our datum line has necessarily introduced, may be left 

 out of account in our dynamical problem. 



If the slope of the ridge is not an infinitely small fraction 

 of a radian our solution will still hold, provided its height is 

 very small in comparison with the depth of the water over it. 

 But the effective potency of the ridge would then not be its 

 profile-area A, but something much greater; of which the 

 amount would be found by taking a stream-line over it, far 

 enough above it to have nowhere more than an infinitesimal 

 slope, and finding the profile-area of such a stream-line above 

 its own average level considered as the virtual bottom. With 

 these explanations we shall speak of a ridge for brevity instead 

 of an " irregularity " or " obstacle," and call its profile-area A, 

 simply the " magnitude of the ridge ;" this being, as we see 

 by (15), the measure of its potency in disturbing the surface. 

 When instead of a ridge we have a hollow, A is negative ; 

 and when convenient we may, of course, call a hollow a nega- 

 tive ridge. 



It is clear that (15) converges, and does not depend for its 

 convergence on k being less than unity ; so that in it we may 

 take k absolutely equal to unity, and we shall do so accordingly. 



