STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 329 
—for the smoothest possible channel—and ‘055 fora channel with 
irregular banks, covered with aquatic plants. Not one of these 
formule recognises the influence of temperature. 
For a pipe subject to flow at constant temperature but under 
different pressures, we have from (14) and (15), since the radius, 
and the roughness of the boundary, are constants, 
= Kk i 
k denoting k’ R? in the Chezy formula, and #?/(a #+/) in the 
Darcy and Bazin; that is to say k is a quantity which does not 
vary with either U or 7. Hence this expression affirms that the 
rate of fall of presswre—that is the slope, or hydraulic gradient— 
varies as the square of the velocity. 
In Kutter’s formula, as it is generally called, let us put 
y+tb=A;ay+ VR=4p; andcy=v 
so that A, p a v are susiotaly constants in the case of any pipe, 
then we shall have from (16) 
LXpy\3 
U2 = R? (Tas) I (18) 
which implies that the hydraulic gradient varies as the square of 
the velocity only when A = p, that is when VR =, or expressed 
in centimetres, is 10. It is obvious that any sensible deviation 
from the law expressed by either of these formule, viz. (17) or 
(18), is a sufficient reason for rejecting them as empirical express- 
ions representing the relation of the mean velocity to the rate of 
fall in pressure. Further it will be quite unnecessary to discuss 
the latter somewhat complex relation, if it be shewn that UN« i, 
m being a simple index. It may also be noticed that it would be 
easy to examine whether in the case of a pipe of 100 cm. ‘hydraulic 
radius,’ or 200 em. actual radius, the velocity varied as the square 
of the hydraulic gradient. Ganguillet and Kutter lay great stress 
upon their discovery of this relation, which implies that the index 
of U varies with the radius, though this consequence was not 
Specifically indicated by them, inasmuch as, accepting the express- 
ion k' vV(RI ), they concerned themselves = with the law of 
variation of k’, 
