156 



EXAMPLE. 



87. The Time of Emptying any Vessel through 

 a Vertical Orifice. Let A be the surface 

 of the liquid in the vessel when the orifice 

 OCD is opened, and H the surface at the end 

 of the time t ; let AH = z, AO = h', OE 

 = .T, AB = h, and PQ, = 2y. 



Then the quantity discharged through the 

 orifice in an element of time, from (9) of 

 Art. 86, is 



Q ~ 



^zdx \dt, 



the ar-integration being taken between h h' and 0, z 

 being constant during this integration ; and since, in the 

 same time, the surface of the liquid at H descends a dis- 

 tance dz, the quantity discharged through the orifice in 

 this time must equal K dz, where K is the area of the sec- 

 tion of the vessel at H. Hence, we have 



2\/2gJ yVx + h' zdx \dt = Kdz\ 

 " ZVtyJ , 



fyVx + h' zdx' 

 the ^-integration being taken between and h. 



EXAMPLE. 



Find the time of emptying a cone 

 by an orifice ACB in its side. 



Let AH = h be the axis of the cone, 

 CB = ft, CA = 7, angle HAG = , 

 AK = x, PK being perpendicular to 

 AH. When the orifice is opened, let 

 the surface of the liquid in the vessel 

 be at H, and at the end of the time t 

 let it be at M, and let AM = z. 



(3) 



