4 S. YOKOTA: A METHOD OF DETERMINING THE CROSS-SECTION 
(b) Cross Section of Variable Depths deduced from that of 
minimum wetted perimeter. 
Although the cross section of minimum wetted perimeter is the 
most economical from a theoretical point of view, it often happens that 
it is desirable on several practical considerations to have shallower or 
deeper sections than that determined theoretically. 
For this purpose, we may take the cross section of minimum wetted 
perimeter as the standard and investigate the variations of its sectional 
area and wetted perimeter when it is changed into any arbitrary trape- 
zoidal section while maintaining the same constant discharge, 
The result of my investigation shows that, if we take for ordinate 
the ratio m (= i of the arbitrary depth to the standard and for abs- 
cissa the corresponding ratio x (= By of the bottom breadths, the curve 
for a constant discharge is a variable hyperbola, and varies very slightly 
for a constant inclination @ (See Appendix). 
These hyperbolas for @ =30°, 45°, 60° and 90° are plotted in Plates 
I to IV respectively. 
The ratios of the depths and the breadths referred to the standard 
section being thus given by a point on the curve, the cross section of 
any arbitrary depth may be determined as in the following examples. 
Conversely, given a cross section, J & n, to find the Q or the velocity 
of flow, we may proceed as follows :— 
We have 
Mis Oe eb 
“== s=—1, —, a known number, say F. 
X b, b, d / d y 
Therefore, at x==1, take m=F and join this point with the origin. 
The point of intersection of this line with the hyperbola is the correspond- 
ing point of the given section, and by dividing b, by the corresponding 
x, we get the bottom breadth of the standard section, whence Q_ is 
read off from the curve (4). 
This Q divided by the given sectional area is the mean velocity 
required, 
