STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 351 
case of the general formula (40) is no better. It is impossible, in 
view of the results shewn in Figs. 2 to 6, to escape from the 
recognition of the difficulty of estimating the roughness; and 
evidently, even under apparently identical conditions, it has 
sensibly different values. Hence all practical computations of 
velocity are subject to a considerable margin of doubt ; for which 
there appears to be no remedy. 
20. The so-called hydraulic-radius.—In hydraulic formule it is 
generally assumed that the resistance to flow, propagated from the 
wetted surface of a pipe or channel, may be always expressed as 
a function of the hydraulic-radius merely—i.e., as the quotient 
formed by dividing the area of a right section by the wetted 
perimeter, a quantity which we shall denote by R or r—and that 
in this way the flow in any form of pipe (or channel) is i liatel 
comparable with the flow in a circular pipe. If the use of the 
hydraulic-radius wholly eliminated the influence of the form of 
the channel, the assumption would be entirely satisfactory: but 
such is not the case. 
21. Corrected hydraulic-radius for ellipse: rectilinear flow.— 
With rectilinear flow in a pipe of elliptical section, the semiaxes 
being B and C, the variation of velocity with size of pipe—all 
other circumstances remaining the same—may be expressed by 
kU = B*O*/} (B* + C*). = £ aay.....csees. (42 
The hydraulic-radius analogue of this quantity, S denoting the 
area of the ellipse and Z its circumference, is:— 
(8/2#)? = B*C2/(} B-4./BC +20)? = x say oe. eeeeeee (43) 
this last expression being exact up to and inclusive of the sixth 
power of the excentricity of the ellipse.’ 
Puttin 
ing 
R= 3 (B+C) and «<=(B-C)/(B+C) (44) 
so that 
B= R(1+6) and C= R (1-98), (45) 
Pipe, Bor use these letters to distinguish the quantity from the radius of a 
2 The term is nag For this approximati Boussinesq, Comptes 
Tendus, t. 108, pp. 695 - 699. 
