Comparing their own measurements in torrential streams in Switzerland with those of Darcy and Bazin, Ganguillet and Kutter found that the four classes of coefficients proposed by Darcy and Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an additional class was required for channels obstructed by detritus.
To obtain a new expression for the coefficient in the formula
v = √ (2g / ζ) √ (mi) = c √ (mi),
Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form
c = α / (1 + β/√ m)
could be made to fit the experiments somewhat better than Darcy’s expression. Inverting this, we get
1/c = 1/α + β/α √ m,
an equation to a straight line having 1/√m for abscissa, 1/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is β/α.
Plotting the experimental values of 1/c and 1/√ m, the points so found indicated a curved rather than a straight line, so that β must depend on α. After much comparison the following form was arrived at—
c = (A + l/n) / (1 + An / √ m),