The fundamental formula for calculating the velocity of water flowing through a pipe or channel, and for calculating the rate of discharge, is based on that of Chezy, a French engineer, who proposed in 1775 the formula:

"Where .

V = mean velocity of water in feet per second. R = hydraulic mean depth = area in sq. ft. of cross-section wetted perimeter in feet S = slope = inclination of water surface length of pipe or channel c = a coefficient determined by experiment and fixed by Chezy at 93.4. This formula was further investigated by Eytelwein, a German experimenter, between 1814-15, and slightly modified, but his form gives practically the same result. The next important formula proposed was that by Neville in the middle of the century, giving a different value for the coefficient c from that of the earlier experimenters. About this time Weisbach introduced his well-known formula, which has been for the last, thirty years so much used by hydraulic engiueers in this country; it is more complicated than any previous one, a varying coefficient c being given, depending on the rate of the velocity. From 18.50 to 1858 M. H. Darcy began in France a remarkable series of experiments on open channels and pipes, on a much larger scale than had previously been attempted.

Darcy died in 1858, and his work was continued by his assistant, M. H. Bazin. The latest, and by far the most important, researches on the flow of water are due to Gauguillet and Kutter, of Berne, who published their researches in 1869 and 1870. These experimenters continued on the lines of Darcy and Bazin, and found that the Chezy formula could be adapted to all cases, but that the value of the coefficient c varies under very many conditions instead of remaining constant, as in the early form. Kutter established a series of " coefficients of roughness " which have been largely experimented upon in America, Germany, and England, and have been proved to be substantially accurate. The following table shows more clearly the great difference between different formulae. Comparison of formulae: -

New formulae proposed are either modifications of the Darcy and Bazin or Kutter forms, or, being dependent upon a single isolated experiment, are not entitled to any authority.