STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 315 
1. Introduction.—Though the problem of the motion of water 
in pipes and channels, said by St. Venant to constitute a hopeless 
enigma,’ has received some attention from mathematical physicists, 
even its empirical solution cannot yet be said to have been satis- 
factorily reached. The formule used by engineers are in general 
either those furnished by MM. Darcy and Bazin, or the very 
ingenious modification by which MM. Ganguillet and Kutter 
endeavoured to embrace every possible case of flow in uniform 
channels. The degree of precision to which results, founded upon 
such formule, are usually expressed, indicate how imperfectly 
their limitations are appreciated: it will be shewn in the course 
of this paper, that even in respect of their mathematical form 
they are systematically defective. 
2. The motion of water in a pipe or channel.—In a pipe or 
channel of uniform section the steady motion of water—defined 
by the condition du/dt =0—*under the action of an accelerating 
force such as gravity, involves the recognition of an equal and 
equally constant retardation ; which must be conceived as arising 
from the internal friction of the fluid, in some cases largely 
influenced, however, as to the mode of its action, by boundary 
conditions, In a uniform and horizontal pipe there is always a 
fallin pressure from point to point, in the direction of the motion 
of the fluid within, due to the internal friction. 
The part played by friction was perhaps first clearly recognized 
by Mariotte in 1686,and following him, by Guglielmini, Couplet, 
D’Alembert, Bossut, and DuBuat. Nevertheless it was not till 
1799 that a satisfactory attempt was made to obtain a numerical 
expression for it. The course of the investigation of this quantity 
—the viscosity constant for water see with a reéxamination 
1 Déspérante 6 énigme.—C. R. t. 74, p. 774. 
2 In which as usual, ¢ denotes time and u denotes the Jalesies parallel 
to the axis of any point in a right section; the actual velocity of each 
particle, when the motion is rectilinear, or the mean of the velocities — 
when non-linear. The latter condition is in some sense peri ic. 
3 Traité du mouvement des eaux. Paris 1686. 
