Dr. Young’s Hydraulic Investigations. 183 
zontal; but where v is small, y may be taken, without material 
error, m y/ or even m </ ti, which is the velocity of every 
small wave. The coefficient m is here assumed the same for 
the motion of a wave, as for the discharge through an aper- 
ture, and I have reason from observation to think this estima- 
tion 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 
of pressure, occasioned by the elevation before it, and the de- 
pression behind, will cause the fluid, immediately below the 
moveable plane, to flow backwards, with the velocity deter- 
mined by the height, which is the difference between the le- 
vels ; 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 depression, 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 back- 
wards without limit : in the first case, the elevation % will be 
to the depression as v to y — v, the length of the same por- 
tion of the fluid being varied inversely in that ratio ; in the 
second case, the proportion will be as y -f- v toy — v : and 
the difference of the levels will be % —ti or second- 
ly z -J- z and first, m y/ ^ c -j- (y — z >}z = ( a — c) 
v ; but, since y is here considered as equal to m y/ ”, putting 
'L — y/ b = d, y — v = md, and, calling a — c, e, m y/ ~ 
c -j- mdz — me </ b, ^ c -f- dz — e y/ b, t® ~ = e r b - f- d*z t 
— 2 dze V — and, calling + 
