222 TRAPEZOIDAL SECTION OF LEAST RESISTANCE. 



ABCD is the one which causes the least resistance of fric- 

 tion, and therefore is the best for open channels. Also, of 

 all trapezoidal sections, the semi-hexagon ABCD is the one 

 which causes the least resistance of friction ; and so on to 

 the other cases. But the semicircle will present less resist- 

 ance of friction than the semi-hexagon, and this latter less 

 than the semi-square. The half decagon offers still less 

 resistance than the half hexagon or the half square. The 

 circular and square sections are used only for troughs 

 made of iron, stone, or wood. The trapezoid is employed in 

 canals, which are dug out or walled up. It is very rare that 

 other forms are used, owing to the difficulty of constructing 

 them. 



120. Trapezoidal Section of a Canal of Least 

 Resistance, when the Slope of the Sides is Given. 



Let ABCD be the section. Put 



x = AB, the width of the bottom, ^J? f 4_^ 



y BE, the depth, and = BCE, 



the angle of the slope, which is to 



be considered as a given quantity, 



dependent upon the nature of the 



ground in which the canal is excavated, and a = the given 



area of the section ABCD. 



Then the wetted perimeter of the section is 



p = AB -f SBC 

 = x + 2y cosec 0. (1) 



The area of the section is 



a = xy -f- y* cot : 



which in (1) gives 



_ a / 2 cot 6 



X ; 



y 



a V* cot 6 /^ 



P = - - + %y cosec 0. (6) 



