352 G. H. KNIBBS. 
the ratio of the former expression to the latter, becomes identically 
& R* (1 — 3c? +4e* — 4e°...) 
x RB (1-fe? +t e* -He’...) 
Consequently this last quantity is a correcting factor to be applied 
to the square of the hydraulic radius for the case of rectilinear 
flow, or taking its square root the hydraulic radius R, of an ellipse, 
=l1-fe +i ct —He' etc. (46) 
requires to be multiplied as in the phering expression 
R, = R(l-Fe +x & —a% e*...)......... (47 
R, denoting what might be called the corrected hydraulic radius. 
It is evident from these last equations that different forms of 
channels are not comparable unless some correction be applied to 
the hydraulic radius, or to express this otherwise,—the simple 
function called the hydraulic radius is not adequate, when precision 
of a high order is required. If there be a marked departure from 
the circular form, the hydraulic radius must be modified. e 
following table will perhaps more clearly illustrate the significance 
of this statement. 
Table VI. 
Values of « in the correction, R, = R (1 +2) for pipes of elliptical 
section. 
Velne wld) 05.5 1055.15, 20. 326-9 30... 85. 100 
‘iy —-0006 -0025 -0055 -0095 -0144 -0200 -0258 -0320 
22. On the investigation of the law of flow in channels.—Un- 
fortunately the magnificent series of experiments made by Bazin 
on flow in channels, were with a few exceptions, made with rec- 
tangular instead of with —— oe! § eae, the 
results for different hydrauli liately comparable, 
inasmuch—as is evident from the preceding section—the unknow? 
correction to the hydraulic radius systematically changes through 
out any series of experiments. The only exception to this is Series 
23, with a wooden triangular channel, and these, made with the 
one slope, permit of the law of variation of velocity with increase 
of hydraulic radius being determined. In order to find the law of 
slope, it is necessary to have experiments in which the radius is 
kept constant and the slope varied, These can only be obtained 
from Bazin’s experiments by interpolations. 
