*ss Hydraulic Investigations. 



the velocity would be ?/, is nearly a + f */ ( c 2al) : conse- 

 quently, when the velocity v is at all considerable, y must 

 be somewhat greater than the velocity of a wave moving on 

 the surface of the elevated fluid ; and probably the surface 

 ©f the elevated portion will not in this case be perfectly ho- 

 rizontal ; but where v is imall, y may be taken, without 



material error) m~ *j —, oneven m j -^-, which is the velo- 

 city of every small wave* The coefficient m is here assumed 

 the same for the motion of a wave, as for the discharge 

 through an aperture, and I have reason from observation to 

 think this estimation sufficiently correct. 



Supposing now the moveable end of the canal to remain 

 open at the lower part as far as the height c, then the excess 

 bt" pressure, occasioned by the elevation before it, and the 

 depression behind, will cause the fluid, immediately below 

 the moveable plane, to flow backwards, with the velocity 

 determined by the height, which is the difference between 

 the levels; and the quantity thus flowing back, together 

 with that which is contained in the moveable elevation, must 

 be equal to the whole quantity displaced. But the depres- 

 sion, behind the moveable body, must vary according to the 

 circumstances of the canal, whether it be supposed to end 

 abruptly at 'the part from which the motion begins, or to be 

 continued backwards without limit : in the first case, the 

 elevation z will be to the depression as v\oy — v, the length 

 of the same portion of the fluid being varied inversely in 

 that ratio; in the second case, the proportion will be as 

 y -f- v to y — v : and the difference of the levels will be 



y — v zy ., y—v 2zy . r 



z 4- z = — i or secondly z -f z ■-■ -- = —f—i and first, 



v v y-\~ v y-\~ v 



m A / c -J- (y — v) z — (a — c) v; but, since y is here 



considered as equal to m „/ -, putting A / — A / h = d, y 



1 2 



7 1 11- "'I 7 ' 



— v = md i and, calling a — c, c, m ^ -*- c '+ mdz = me 

 V by */ 3 c -f dz = e V h c l — =i e'h + d'z 1 — Qdze j l, 



V v 



- 



z* — 



