﻿Problem of the Weir. 787 



above the upper stream-bed, or 5 ft. above the lower stream- 

 bed, the hydrostatic pressure would be about 280 lb. wt. 

 per foot breadth of weir. The effect of the high stream 

 velocity chosen is to reduce the pressure on the weir to about 

 one half of its static value. 



Using the same values of U, /i, and cl, the following are 

 the stream velocities along the lower bed: — 18*978, 3*1232, 

 and 4*4768 ft. per sec. respectively. It appears, therefore, 

 that the case II. realizes most nearly the condition of no 

 slipping along the lower boundary of the stream. 



§ 14. Conclusion. 



The obstruction over which the water flows has been 

 alluded to, for shortness, as a weir, but the exact form which 

 this should take, in order to secure the conditions essential 

 to the theory, is not clear. If the flow over the fall be free 

 from turbulence or discontinuity of any kind the results 

 worked out may serve as rough approximations to what is 

 taking place. They would apply more closely, probably, to 

 the case of a stream flowing over a bed whose slope is every- 

 where gentle. In the latter case Lord Kelvin has stated 

 that corrugations will not appear on the down-stream surface, 

 and the fundamental hypothesis of § 1 will, therefore, be 

 vindicated. In any case the rapidity with which a uniform 

 regime is usually established on passing down-stream from 

 the fall is somewhat remarkable, considering the smallness 

 of viscosity. 



The question of the stability of the various types of motion 

 worked out has been left quite untouched, as the method is 

 evidently unsuitable for its discussion. The results may, 

 however, help to assign maximum or minimum limits to the 

 various quantities involved, and may also serve to indicate 

 some of the difficulties which might be encountered in any 

 attempt at a more complete solution. 



Appendix A. 



It has been stated that in a uniform horizontal canal of 

 rectangular cross-section it is possible for a standing finite 

 elevation of the stream-surface to be permanently maintained 

 when the depths and velocities of the stream on either side 

 of the elevation take up suitable values. The impossibility 

 of this has been virtually implied in the preceding work, and 

 the proof now given covers the most general case of non- 

 viscous motion. , 



3 F 2 



