384 Hydrodynamics. 



we shall have 842,94, differing from the experimental result 

 about / part. 



497. Let it be proposed to find the quantity of air expelled 



Fig. 233. into an infinite void from the aperture C of the vessel ABCD 



during any time , and the density of the remaining air at the 

 end of that time. 



The bulk expelled during the instant d t will be 6 d t \/Zgh) 

 the velocity ^/2 gh being constant, and consequently the quantity 

 will be <* D 7 d t \/ c lgh. The quantity at the beginning of the efflux 

 is b D, 6 being as before the bulk of the vessel ; and when the 

 air has acquired the density D', the quantity in the vessel is b D', 

 and the quantity expelled is 6 D 60'; consequently, the quan- 

 tity discharged during the instant d t must be the differential of 

 5 D 6 D X , that is, b d D'. Hence we have the equation 



6 D' d t \/2 g h = b d 

 and 



_ 



the integral of which is 



h. log. denoting the hyperbolic or Naperian logarithm of r/. 

 When t = 0, D' is equal to D ; whence 



b 

 c = - : -- h. losf. D : 



therefore, 



nearly. 



It is obvious that no finite time will be sufficient for the ves- 

 sel to empty itself; for, as D' must in this case be equal to zero, 



^ will be infinite, and its logarithm will also be infinite ; so that 

 t will be infinite. 



