﻿Problem of the Weir. Ill 



for the present that (7) does not bold, write ox— l-j-77, 

 S— 1 + e in (6). The general solution of (6) will then be of 

 the form 



V=f{<<:-V)}, 



or e^ + Ffo)/*, (8) 



where F(^) is expansible in positive powers of tj and does 

 not contain a constant term. The problem of determing 7] 

 in terms of k and e may be solved by using Lagrange's 

 theorem ; winch gives 



'='+?yr- 5 ^w.)}-. ... • (9)t 



Since F(e) vanishes with e, therefore (9) makes ?;— >0 as 

 e— >0. It also makes rj-^e (i. e., *7->$) when /c— >co . This 

 branch of the function therefore satisfies the two physical 

 conditions mentioned above, and (9) will therefore be taken 

 as the formal solution of the problem. 



In order to justify this selection we note that the value 

 3=1 makes the total fall of the stream-bed zero, and the 

 corresponding value <sr=l makes the stream-surface level. 

 If some other root of (6) had been chosen, then on making 

 5 tend to unity, -37 would not have tended also to unity, 

 since relation (7) is not supposed to hold. This would infer 

 the possibility of a standing elevation of some kind, in a 

 uniform horizontal canal, in which the free surfaces on either 

 side of the elevation were at a finite difference of level. 

 The impossibility of this is proved in Appendix (A) for the 

 most general case of non-viscous flow. 



If equation (7) be satisfied the above argument breaks 

 down, since there will be two values vr=l corresponding to 

 5 = 1, and therefore two expansions for y in terms of e cor- 

 responding to the two branches of the function (?>). It 

 seems probable that the motion is unstable when (7) is true, 

 and is of a different type according as 



Jo 



A particular case is given in § 5. 



§ 1. In the case of a weir it is interesting to determine 

 the resultant horizontal pressure on the face of the weir. 



* Whittaker, ' Modern Analysis,' p. 106. A 



t The series under 2 will contain a term in c if d( ( HO ls ~ con- 

 vergent, and vice versa. In any case 77 ->•<? as k— x . ' ° 



