HYDRAULIC INVESTIGATIONS. JO I 



elevated portion will not in this ca9e by perfectly "horizontal i 



but where v is small, y may be taken, without material er- 



x <i . . 



rour, m \/ — , or even m */ — , 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 aperture, and I have reason from observation to think 

 thjs estimation sufficiently correct. 



Supposing now the movable end of the canal to remain Case where the 

 open at the lower part as far as the height <?, then the ex- moving end re- 

 cess of pressure, occasioned by the elevation before it, and [JSow ° P 

 the depression behind, will cause the fluid, immediately be- 

 low the movable 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 movable elevation, must 

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

 sion, behind the movable 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 to y — v, the length 

 qf the same portion of the fluid being varied inversely ia 

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

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



Vr—v zy ., y — v 2zy i m 



33 — , or secondly z -f- z — — - 33 — — - ; and firit, m */ 



v v y -{■ v y -{• v 



z if 



—- c -{- (y-—v) z S3 [a— -c) v; but, since y is here consi- 

 dered as equal to m \/ — putting */ —--A/b~d > y—-v 



z it 

 S3 m d, and, calling a — c, e, m*/-^-c + mdz — me v' 



v 



b, </-? c + dz~e -v/ b,c % -^ -. ft + d % z % —2dzc V 



V V 



by z 1 — . ( ~-\ - — J z =z — -=• -, and, calling — ;£-. -f 



\a~v d J d z * 2d*v 



-J~ >/>*=/— V (/-^) : anc * in ^ e 9ame planner 



/is 



