﻿782 Mr. J. H. C. Searle on the 



Hence, when tc < 2, i. e., gh ■< U 2 or <i < 7i, it is easy to 

 derive the following equations : — 



«;=l/*=l/5,U / = U(l + i//0, and K'=h*/(h+'dj. 



When « > 2, z. £., (7A >* U 2 or f? >■ h, the corresponding 

 values of -cr, XT', and 7/ are found from the quadratic 



■57 



■ kvt-1 = 0. 



§ 8. To calculate the horizontal pressure on the face of 

 the weir put <£(£) = * m C 11 )- We fincl 



&=7rd + lT 2 h{iK{vj~ + VT)-l}(vT-l)/*T. . (16) 



7ri is the atmospheric pressure on the face of the weir and, 

 by means of the inequalities iff ^ 1 according as k ^ 2, it 

 may be shown from (16) that 2P always > ird, so that the 

 pressure is always positive. £P may be found from (16) 

 when iff has been determined for any special case from (6). 



When /e = 6\ as in § 7, and -ar = 1/k = 1/8, the corresponding 

 pressure may be found at once, viz., 



&=im(8-l){38-l)/S = yVd(2h + M)l(h + d)*. 



§ 9. Turning now to the case where the motion is vortical, 

 let the vorticity of the fluid be everywhere constant. Along 

 both the upper and the lower streams the motion will now 

 be of the "laminar" type given by the ordinary equations 

 when viscosity is retained. Taking the velocity to be zero 

 along the upper stream-bed so as to correspond as nearly as 

 possible to the actual physical conditions, we have 



tt=u.(*-<o/A=u{:. 



Hence, by (5), <p(£) = £ 2 , and from (6), after integration, 



*={l + *(8- : w}*-{ic($-«r)}*, . . . (17) 



from which <s7 4 + 4kot 3 -2ot 2 (1 + 2kB) + 1 = 0. . . (18) 



The roots of (18) are distributed as follows : — 



— cc ■< iff 1 < — 1 •< w 2 < < «r 3 < 1 < tsr 4 <£ $. 



When S=l (18) becomes 



(^-l){^ 3 + (l + l/<) OT 2 -^-l}=0, 



of which the physical solution required is «r=l*. There 

 will not be a second value «r = l unless k=0. 



* Appendix A. 



