﻿784 Mr. J. H. C. Searle on the 



When U, h, and d are given k and & are found" from (1) 

 and, thence, ts may be determined from (17) or (18). This 

 value when substituted in (19) will give the corresponding 

 horizontal pressure on the weir face or sloping bed. 



§ 11. Amongst engineers a formula of the "parabolic" 

 type has been proposed as representing the distribution of 

 velocity across a vertical section of a uniform canal. Thus, 

 the equation u = u -\-/3z-{-yz 2 , in which a is zero or very small, 

 would represent a type of motion in which the velocity is 

 zero, or very nearly so, along the bed of the canal and in- 

 creased upwards. Experience seemed to show that the velocity 

 was a maximum not at the surface but at some point below 

 it not greater than one fourth the stream depth *. 



The adoption of the above formula in the present note 

 would lead to some very unwieldy analysis, and it is there- 

 fore proposed to substitute the formula 



M = Usin{i9r(*-d)/7i}, 



which makes the velocity zero along the upper stream-bed 

 and a maximum at its surface. Hence, 



2« = Usin Wf, or by (5), </>(?) = sm2 "? ?■ 



Substituting in (6) and integrating we find, after some 

 reductions, 



{l + /c(S-^)} sin 2 1^=1. . . . (20) 



[Writing vt = 1 + t) and S=l + e, (20) may be transformed 

 into 



e — ??-}- - tan 2 ^ttt]. 



Using (9), we have 

 X 2 2j _2V 



/5X 6 2X 4 \ . / X 8 x G \ . 



-(**?" + 1 i? +45 r>": + -' 



where X = 7r/2. It may be more advantageous, however, to 

 expand in terms of tan ^7T€.] 



It may be verified that the root required is such that 



1 < Z7 < 8. 



* Bcvey, I. c. ante, p. 220. When the stream width is small com- 

 pared with its depth the point of maximum Telocity may be lower than 

 stated above. 



