370 Hydrodynamics. 



acquire in falling through the same space by the pressure of the 

 column CIKD. If we suppose the lowest stratum ACDB, to 

 fall as a heavy body through the height BD, the moving force 

 will be its own weight. But if we suppose it to be urged by its 

 own weight, together with the pressure of the incumbent column 

 of fluid CIKD through the same space, the velocity in the former 

 case will be to that in the latter, as the moving forces and the 

 times in which they act, the mass moved being the same in 

 both cases. But the moving forces are to each other, as the 

 27. heights BD, KD, and the times in which they act, the space 

 6 ' being the same, are inversely as the velocities. Accordingly, 



BD KD 



v : u : : --- : - . 

 v u 



Whence 



v 2 : u 2 : : BD : KD, 

 or 



v : u :: V#D : VKD. 



Now v is the velocity which a heavy body would actually ac- 

 quire in falling through the space BD, and as the velocities, 

 other things being the same, are as the square roots of the spa- 

 26g ces, u the velocity of the issuing fluid is that which a heavy body 

 would acquire in falling through KD, the height of the fluid above 

 the orifice. Therefore, the velocity with which a fluid is discharg- 

 ed from the bottom of a vessel is equal to that acquired by a heavy 

 body in falling through a space equal to the height of the fluid above 

 the orifice. Also if a pipe Jl'B'C'D' be inserted horizontally, 

 or inclined in any way to the horizon, it may be shown, in 

 like manner, since the pressure of fluids is equal in all directions, 

 that the fluid will be discharged with the same velocity as be- 

 fore. It will accordingly ascend to the level of the fluid in the 

 vessel, all obstructions being removed ; and it is found in fact, 

 under the most favourable circumstances, nearly to reach this 

 point. It follows, moreover, from what is above laid down, that 

 if apertures be made at different distances s, s', s", below the 

 surface, the velocities at these points, and consequently the 

 quantities of fluid discharged at these points, from apertures of 



