Discharge of Fluids through Apertures. 375 



and the quantities of fluid discharged in a given time through 268. 

 indefinitely small parallelograms at these depths are in the same 

 ratio. But the parabolic curve EKH being drawn, having EG 

 for its axis, we have 



EG : El :: GH : IK*, 



and consequently, Trig.l7 



\/EG : VET :: GH : IK\ 



whence the quantities discharged through indefinitely small par- 

 allelograms at the depths EG, El, are to each other as the ordi- 

 nates GH, IK, and the sum of all the quantities discharged 

 through all the parallelograms of which the rectangle EFGH is 

 composed, is to the sum of all the quantities discharged through 

 as many parallelograms at the depth EG, as the sum of all the 

 elements IKk i of the parabola, to the sum of all the correspond- 

 ing elements IL I i of the rectangle ; that is, as the area of the 

 parabola EKHG to the area of the rectangle EFGH-, in other 

 words, the quantity running through the notch EFGH is to the 

 quantity running through an equal horizontal area at the depth 

 EG, as EKHG to EGHF, that is, as 2 to 3. Therefore the cal. 94. 

 mean velocity of the fluid in the notch is equal to two thirds of 

 that at the greatest depth GH. 



485. If a small aperture be made in the side of a vessel kept pi 2g6 

 filled to the same height, the fluid will spout out horizontally with 

 the velocity acquired by a heavy body in falling freely through the 

 height of the fluid above the aperture, and this velocity combin- 477. 

 ed with the perpendicular velocity arising from the action of 

 gravity, will cause each particle, and consequently the whole jet 

 to describe a parabola. Now the velocity with which the fluid 303. 

 is expelled from any aperture, as G, is such as would, if uniform- 

 ly preserved, carry a particle through a space equal to 2 jBG 266- 

 in the time of its natural descent through BG ; accordingly, if the 

 direction of the aperture be horizontal, the action of gravity 

 being at right angles to it will cause the particle to descend 

 through the height GD in the same time that would be required 

 in case of a natural descent through GD, if no other force were 



