304 G. H. KNIBBS. 
hydraulic radius throughout anyone series of slopes. The best 
slopes for the sides would be 1 to 1, so that the angle included 
would be 90°. 
24. Proof that the index of the hydraulic radius varies with the 
radius.—On plotting the logarithms of the velocities and hydraulic 
radii of Series 23, respectively as ordinates and abscisse, to which 
reference has already been made (§ 21), it is quite evident that the 
points lie in a curve, Fig. 10, convex upwards. Hence m’, the 
value of d (log R)/d (log UV) diminishes as Rincreases. This index 
would require to be multiplied by » to be comparable with m in 
the formula for pipes. It is evident therefore that the law of the 
indices, m and n, must be thoroughly ascertained in order to 
deduce anything like an exact expression for velocity of flow in 
open channels. 
25. Dissimilarity of flow with varying hydraulic radius and 
dissimilar forms of channel.—It might be anticipated that when 
the boundary of a channel is very rough, the dissimilarity of the 
law of flow with dissimilarity of form of channel, would be most 
striking. This is well shewn by Bazin’s Series 30 and 31—slopes 
0081 and -0152, breadth of channel (lined with canvas) 10 cm., 
temperature 10° C.—from which the following results are deduced. 
Hydr. Rad. 1:16 169 205 2:39 2°91 
Log U, 1338 1:530 1607 1-662 1-733 
U, 1387 1595 1:694 1-760 1-833 
A log I/ Alog 75-6 4°2 371 2°8 2-7 
These final figures corresponding to the values of n in the investi- 
gation of flow in pipes, indicate that the general formula developed 
for pipes will perhaps fail when applied to open channels. It is 
however likely that an analogous derivation of a formula will be 
practicable: this however I have not yet thoroughly examined. 
26. Indications for further experimental investigation with pipes 
and channels : conclusion.—(1) The law of velocity as related to 
temperature with at least two, better three, pipes of very different 
roughness requires further experimental investigation, especially 
with a view to the determination of the influence of temperature 
