STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 345 
A perfectly analogous method may be followed in regard to m, 
supposing it to vary with RA: case iii, § 16. It may be noticed 
from Fig. 8, that there is a very slight indication of decrease in 
m with increase of R. This may be seen in Table H., and also in 
the following way :—Let the mean be taken of all the small radii, 
i.¢., under 2 cm., and of the correspondent values of m: and 
similarly also of all the larger, i.e, between 5 and 10cm. The 
results are :—for R=1:25, m=1°23 and for R=7°32, m=1-16. 
This decrease with increase of R is distinctly confirmed by the 
position of the point R= 25 em. in Fig. 8, and though, as already 
remarked, there is intrinsic evidence that the precision of this 
Series of observations is not high, it has to be remembered that 
there is confirmatory evidence in the case of open channels, as will 
be shewn in § 24 hereinafter. If then the formula is to be made 
to accord more exactly with the results, shewn in the figure, a 
second-degree curve must be made to pass through the points 
whose radii are 1:34, 4095 and 25-0, and will then well represent 
the whole series. By a rough calculation the tangencies or values 
of m at the extremities and middle thereof are obtained, as shewn 
hereunder. Now in regard to these, the values seem respectively 
rather high and low at the extreme limits, though they are well 
within the range of the results shewn in Table H. If we are to 
have generality, then when & is zero or extremely small, its index — 
must be 2, because the first régime must then exist. Then again 
at the radius, the logarithm of which is the mean of the two 
extreme radii just mentioned, the value of m must be 1-244 (viz., 
for R=5-78). And finally it seems likely that when Z is very 
great its index is unity, the index assigned by St. a ome, These 
conditions will be satisfied ie putting 
pee Ge eben ater (37) 
x + ea 
Tf a be made 0:77, and z be made §, the result will be as follows :— 
ble J ; 
R= 0. 1:34 5:78 — -25°0 ome 
m calculated 2 1400 1243 1133 1. 
m observed? 2 17446 1244 1042 cont Moe 
