﻿Problem of the Weir. 783 



Putting or=l-f ?;, £=l + e in (18) we find to the first 

 order r) = e } so that -sr > 1. The root chosen is, therefore, 

 such that 



1 < «r < g. 



[In this case (17) leads to the equation 



—* + ( 1+ '-rb)V 4,f! 



whence, by (a), 



n (1 11 9 12 6\ , 



From (18) <sr may be expressed as follows : — ■ 



*=*(/-!)+ {^V-iy-v^)}* 5 



where fjb 2 — X 2 l(\ 2 — l) and A, is that root of the cubic 

 (A. 2 — 1)(\— 2kS — l) = 2/c 2 lying between 1 and oo . 



When ot has been found from (18) v is most easily calcu- 

 lated from the equation 











2v=ar+l/<sr ? 



whence v 



always 



> 



1. 





Since 









1 < «■ < 8, 



therefore 









l^h'/h^ 1+d/h, 



or 









h< 7i'< 7* + d. 



Hence, although the lower stream is always deeper than 

 the upper, the stream-surface is always depressed on passing 

 from the upper to the lower reaches. 



§ 10. The calculation of the pressure on the weir face for 

 this type of motion is effected by putting $;£*) = J 2 in (11). 

 We find 



^-7r^ + U 2 A[i/c(^ 3 -l) + i-{l 4f c(S- OT )}^ 



-*{*(*-»)}*-*], .... (19) 



which may be rationalized by (17) if necessary. 



Since, in this type of motion, m > 1 and since (l + .r) p - is 

 always greater than l-f# s / 3 , (19) shows that &> is always 

 > ird. 



