104 



Prof. H. Hennessy. Hydraulic Problems on [Apr. 19, 



diameter, so that for a pipe of 2 feet internal diameter the greatest 

 hydraulic mean depth would be when the surface of the liquid had 

 fallen below the top by 4'4928, or nearly 4J- inches. As the velocity 

 of the liquid is nearly as the square root of the hydraulic mean depth, 

 the pipe filled to this height would carry liquid with a velocity 

 slightly greater than when completely full. This conclusion is only 

 true when the effective head of liquid is due solely to the inclination of 

 the pipe. When the level of the liquid within the pipe falls the 

 hydraulic mean depth tends towards its minimum value, and its 

 decrease becomes rapid as the arc diminishes ; thus if is a very small 

 angle 



i A sin 6\ r02 / 0B\ 



But r = L/tt, where L is the length of an arc of a semicircle; 

 hence if the 4th power of is negligible we have u = L# 2 /127r. 



Although pipes and conduits for water supply are usually quite full, 

 those for drainage purposes are most commonly only partly filled with 

 liquid, and the amount of liquid is liable to great fluctuations. This 

 has led to the adoption for drainage pipes of an oval curve for the 

 outline of cross-section, with the longer axis of the oval vertical and 

 terminated at bottom by an arc of greater curvature than at top. 

 The form of this cross- section suggests an inquiry as to how far a 

 curve which has been often treated in isoperimetrical problems would 

 satisfy the conditions for giving a favourable hydraulic mean depth 

 in an open channel with fluctuating contents. We have seen that a 

 particular arc of a circle gives a maximum for the quotient of the 

 area of the segment divided by the perimeter of the arc, and we shall 

 find that there is a particular catenary which gives a maximum for a 

 corresponding quotient of the area included between its perimeter and 

 its chord. 



If as usual we make the directrix the axis of x, a the parameter, 

 and I the length of the curve, then adopting the usual notation 



x = a log ^ + ^fa 8 ~ and y = J(P + 



» 



but in this case, as the area whose quotient divided by the perimeter 

 is to be a maximum is the difference between the rectangle under the 

 coordinates x and y and the area included between the curve, its 

 parameter, and the directrix, we have manifestly — 



xy — fydx 



u- f , 



and as fydx = al, this may be written 



