NEWTON S PRIKCIPIA. 383 



section, always remain in one. This is called the hypo 

 thesis of parallel sections. Supposing the fluid to begin 

 to move from rest, it can be shown that the motion, what 

 ever be the form of the vessel, soon becomes steady. So 

 that we may in general use the preceding formula. 



If the vessel be allowed to empty itself, the solution 

 can still be found on the same supposition. If z be the 

 height of the surface of the water in a cylindrical vessel 

 above the vena contracta, and h be the initial value of z 

 when the motion is supposed to begin from rest, 



1-2? 



2 TC 2 hi 



12 I 



If B be very small, this expression becomes very nearly 



V 2 = 2 g z. 



If some time has elapsed, so that j is a small fraction, 

 we have 



The velocity being then very small, the quantity of 

 liquid that has run out of the vessel in the time t will be 

 very nearly 



if the orifice be an aperture in a very thin plate, the value 



62 

 of B, the area of the vena contracta is about -r of the 



area of the orifice. Calling this area a, the expenditure 

 will be nearly* 



* Poisson, Hydrodynamique, vol. ii. 



