CYLINDRICAL VESSEL WITH TWO SMALL ORIFICES. 145 



81. The Time of Emptying a Paraboloid Let 



the vessel be a paraboloid of revolution round the 

 vertical axis, h its height, and %p its parameter. Then 

 if x is the depth of the orifice in the bottom below the 

 upper surface of the liquid, we have 



which, in (2) of Art. 79, gives 



t- --J^L f x _^ = _ _ =a . + G 



since when x = h, t = 0. 



Making x = 0, and putting r = the radius of the base, 

 (1) becomes 



t - * **"** (2) 



" 



which is the time of emptying the vessel. 



82. Cylindrical Vessel with Two Small Orifices. 



A cylindrical vessel of given dimensions, is filled with 

 a liquid ; there are two given and equal small orifices, 

 one at the bottom, the other bisecting the altitude ; 

 to find the time of emptying the upper half, suppos- 

 ing both orifices to be opened at the same instant. 



Let 2a = the altitude of the vessel, x = the altitude of 

 the surface of the liquid from the upper orifice at the time 

 t, and r = the radius of the base. Then the quantities of 

 liquid which flow through the upper and lower orifices in 

 one second are, respectively, k Vfyz and Tc Vfy (x + a), 

 which in (1) of Art. 79, gives 



