1888.] 



the Cross-sections of Pipes and Channels. 



103 



do ~ 2 \ 



sin — cos N 



W = ^[( e3 -2)Bin e + 2^o se ]. 



When ^ = 0, 6 = tan0, 



this may be satisfied either by = 0, or by some arc between ir and 

 %r. The root = 0, substituted in the value of d 2 u/d0 2 , makes this 

 positive and equal to -Jr, as may be easily shown by expanding sin 

 and cos 0. Let now = tt + /3, and by successive trials we shall 

 find that /3 = 77° 27' nearly satisfies the equation ir + /3 = tan /3. 

 With this value 6> is 257° 27', cos and sin are both negative and 

 d 2 u/d0 2 is also negative, showing that the result gives a maximum for u, 

 which in this case becomes 



u — 1 r (1 + 0-21722) = 0*6086 r, nearly. 



The hydraulic mean depth of a full pipe or of a half-f ull pipe of 

 circular section is 0'5 r, hence that for a section less by about three- 

 tenths of the perimeter of the circle is greater. The area of the 

 section of greatest hydraulic mean depth is 2*74142 r 2 or 0*87169 of the 

 entire circle. If the pipe is nearly horizontal the quantity of liquid 

 contained in it is proportional to the cross-section, hence a circular 

 pipe under such condition has the greatest hydraulic mean depth 

 when it is nearly seven-eighths full; or when the liquid has fallen from 

 the full state so as to have its free surface AB the chord of an arc of 

 102° 33'. The versed sine of this arc is 0*1872 D nearly, D being the 



Fig. 1. 



