Discharge of Fluids through Apertures. 373 



481. If a; denote the space through which the upper surface 

 A descends in the time f, the velocity of the discharged fluid, 

 represented by V%g ( s #)> will vary continually, but may be 

 considered as constant during the indefinitely small time d t ; so 

 that in this time there will escape through the orifice, a prism 

 of the fluid having the area <J of this orifice for its base, and 

 \/2g (s x) for its altitude. Thus the quantity discharged dur- 

 ing the instant d t is 6 dt v/2g (s #). But during the same 

 time the upper surface has descended through the space d x, and 

 the vessel has lost a prism or cylinder of the fluid, whose base is 

 A, and altitude dx, and whose bulk or volume is A do?, whence 



and 



A dx 



As the area A will be given in functions of a?, by the form of 

 the vessel, the second member of this equation may be consider- 

 ed as containing only the variable quantity #, and it will be easy 

 in most cases, by simply integrating, to discover the successive 

 depressions of the surface, and the discharges of the fluid, from 

 any vessel of a known form. 



482. Let the vessel, for example, be an upright prism or cyl- 

 inder ; A in this case will be constant, and we shall have 



dx SA - , r 



- = ----- =^\/sx -j- ^. 



V* x 6/Z 



Now when the time t is 0, the depression of the upper surface 

 A is also ; thus we have at the same time x = 0, and t = ; 

 this condition determines the constant quantity C to be 



and gives for the time of depressing the upper surface through 

 the space a?, as follows, 



