102 Prof. H. Hennessy. Hydraulic Problems on [Apr. I9 r 



^yda 



JV{'+(2)"} 



where the limits of the integrals are taken between the same points 

 on the curve. 



If Z = /da; \/ 1 1 -f (^-^j | is given, then the problem is to find the 



curve which makes fydx a maximum for the given value of I. This is 

 a well-known isoperimetrical problem*, for by the principles of the 

 Calculus of Variations we have in this case — 



dx = 0, 



where a is arbitrary, and therefore 



v 



a dx 

 which gives 



1 d i dyjdx v _ q 



ay i ax \ 



dyjdx a dy 



and y — c' = — (x — c) 2 } the equation of a circle with 



radius = a. This result proves that for a full pipe the circle gives 

 the greatest hydraulic mean depth, but it does not tell what is the 

 particular arc of a circle which gives the greatest quotient for the 

 area of the segment between itself and its chord divided by itself. 

 This is best done by the ordinary methods of maxima and minima as 

 follows : — 



Let 6 represent the angle subtended at centre by the segment of the 

 circle whose radius in r, then — 



= >( 1 - 5 ^) 



* In his ' History of the Calculus of Variations,' p. 69, Todhunter has made a 

 remark on this problem ; namely, that if the curve instead of being closed were 

 required to pass through two given fixed points with the arc between these points 

 of a given length, the constants of integration would not be arbitrary, and there 

 would be two equations from the fact of the circle passing through the given points 

 and another arising from the given length. The solution here given avoids the 

 necessity of two such equations by employing the well-known properties of an arc 

 of a circle and its included segment. — March 29, 1888. 



