1888.] Discharge tlirougli a Pipe of circular Section. 147 



If the pipe was quite full = 2 TT, sin0 = 0, and Q 1 = 2K?r, but for 

 the maximum value of Q we have 



37851 + 0-61794)8 1 * 



_ TT / (5-3785 

 n 5- 



337851 J 



Hence 



Q 1 27T L 5-33785 

 Qa-Q^ 0-00768 Qi. 



The difference thus exceeds | per cent, for the pipe which is filled 

 up to the segment of 308 10'. The supplemental arc being 51 50', 

 it is easy to see that the maximum discharge would occur when the 

 liquid falls below the summit of the inner surface of the pipe by 

 about the twentieth of the diameter. This result might be called a 

 hydraulic paradox, or the condition of a pipe carrying liquid at a 

 small inclination giving a greater discharge when filled up to 

 nineteen-twentieths of its diameter than when completely full. 



Note added December 19, 1888. 



[The hydraulic paradox here referred to as a deduction from the 

 expression for hydraulic mean depth is not so practically important 

 as the question of velocity of the liquid passing through the section of 

 greatest hydraulic mean depth. The maximum hydraulic mean depth 

 for the pipe was found to be 0"6086r, while it is 0'5r for a full pipe. 

 As the velocities may be taken as very approximately proportional to 

 the square roots of the hydraulic mean depths, we shall have for v' t 

 the maximum velocity, 



= l-1033v. 



Or the velocity for the maximum hydraulic depth exceeds the velocity 

 for a full pipe under the conditions specified by 10^ per cent. 



This result may possibly be utilised in circular drain-pipes liable to 

 be coated with deposits.] 



VOL. XLV. 



