﻿776 Mr. J. H. C. Searle on the 



§ 3. Equation (6) will, in general, contain a number o£ 

 radicals after integration, and in the process of rationalizing 

 and simplification, necessary for numerical computation, a 

 number of roots will appear which are irrelevant to the 

 present purpose. After discarding the negative roots there 

 still remains the problem of picking out one from the re- 

 mainder which will adequately satisfy the physical require- 

 ments. Two conditions will suffice to determine this root: — ■ 



(1) As the fall of the stream-bed tends to zero the surface- 

 level of the stream must become uniform; 



(2) As the velocity of the upper stream tends to zero the 

 surface-level of the whole stream must become uniform. 



Expressed analytically these conditions are as follows : — 



(1) As 8 -> 1, ot must also -> 1; 



(2) As k -*■ co , -57 must — > 8. 



It w 7 ill now be shown that there is a solution satisfying 

 both these conditions, and that this solution is, in general, 

 unique. In (6) write 



-37=1 + 77, </>//<: = \, 5 = 1. 

 Hence, 



l + v =\\ij(\- v )idZ, 



= f{\i-(\- v )i}/(\ 



jo 



Jo 



dy(x- v )i{\s+(\- v y 



Hence, ^ = 0, i.e., -uj=1 is a solution when 8 = 1. 



Also, i rfgjf(A-^)*{A*+(\— 1/)*} = lis a solution when 8=1. 



Jo 



A further value 77 = will not satisfy the last relation 

 unless 



( \/f/2X = l, i.e., unless \ } dS/4>(Q = lJ*/gJi 9 



Jo c/0 



i.e., unless C h . , _ . 



9) dz/u 2 =l (7) 



€, 



Hence, when 8 = 1 equation (6) will, in general, furnish 

 one value ^=1 (or rj = 0), and one such value only. If, 

 however, the motion is such that (7) is identically satisfied, 

 then -5i= 1 will be a double root of (6) when 5=1. Assuming 



