1888.] 



the Cross-sections of Pipes and Channels. 



107 



= a [cosec . log tan-|> (V -f 6) — 1] 



= a [2 cosec <9 (tan J0 + j tan 3 10 + &c. - 1) 



= •[0 + 3+ )(*" + fi)- 1 ] 

 ~ T ' 



when 3 and higher powers are omitted ; and remembering that 

 a = Z/(2 - 4) = L/(4'8), we may write for snch an arc — 



u = L0 2 /(19-2). 



An arc of the semicircle at its base subtending the angle 6 has 

 when is small the value L0 2 /12tt, as already pointed out. Hence 

 for a circular channel and for one formed by a catenary of equal peri- 

 meter and maximum area, the hydraulic mean depth for small seg- 

 ments subtending equal angles would be greater for the latter. On 

 looking at the outline of such a catenary inscribed in a semicircle, 

 this result seems to be confirmed, and the curve approaches the 

 oval which experience has led engineers to adopt for the section of 

 pipes carrying fluctuating quantities of liquid. 



The general result of the preceding inquiry may be summed up 

 in the following conclusions : — For all pipes and conduits employed 

 to convey liquid for consumption or for milling power, the circular 

 section is the best, as the level of the liquid in the pipe is rarely, if 

 ever, below half the diameter. 



For drainage such a form is also the best if the liquid rarely falls 

 below half the diameter, but if it is liable to fall nearly to the bottom 

 of the pipe or conduit, an oval form, such as that actually recommended, 

 is the best. If the pipe is likely to be as often half full as slightly 

 filled, it is probable that some advantage would be gained by employ- 

 ing the catenary of maximum area for a given perimeter for the lower 

 part of the oval. A pattern for this form can be always readily con- 

 structed by remembering the relations 1, 2, 3 for the depth, the chord, 

 and the length of the curve. In designing the base of the pipe, it is 

 only necessary, as 'already pointed out, to hang a fine chain of 3 units 

 between supports placed at 2 units on the same horizontal line. 



It is well known that in a triangular notch or triangular channel, 

 the sides of which are at right angles, the velocity of the liquid varies 

 but little with the depth, and it is possible to conceive that a channel 

 may have such a form as to make such a variation extremely small. 



If we suppose the surface of the liquid in an open channel to be 

 bounded by the chord of the cross-section of the channel, then we 

 shall have as before the hydraulic mean depth — 



