108 Problems on the Cross-sections of Pipes, Sfc. [Apr. 19, 



— ' xy — fv dx 



and if we make u = constant — 



•2.K/{'M2y}-'-'-W{' ♦(»'}■ 



the limits of the integrals in both cases being taken on the same 

 points of the curve. 



From this it follows that — 



■2- VI' *(»'}■ 



which on integrating gives 



y = c log {x + v/(* 2 - c*)} + C. . 



This result indicates a catenary with its convexity turned to the 

 chord and to the axis of y, but between the limits x = and x — x 

 the value of y becomes imaginary, the constant c being the hydraulic 

 mean depth, which must be very small in such a case as here sup- 

 posed, if we take x from x = c to x = x 



y = Clog^ V - > 



and such a notch or channel might be approximately realised by two 

 arcs of a catenary with parameters corresponding to the small arbi- 

 trary value of c. 



Fig. 3. 



A C 



R 







I 



) 



A notch or channel with such a cross-section would have an almost 

 constant hydraulic mean depth, but it would be inapplicable to any 

 useful purposes in the application of hydraulics. 



The cross- sections of rivers and navigable canals are regarded 

 chiefly with reference to permanence, and the question of their 

 hydraulic mean depth is less important than in the case of water 



