STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 347 
therefore will not systematically deviate from the relations sub- 
sisting between them. The geometry of this analysis, by means 
of which the preceding formule have been deduced, will make 
obvious what is meant by this statement. The defects of the 
Chezy, of the Darcy and Bazin, of the Ganguillet and Kutter, and 
of the Reynolds’ formula, is that each systematically departs from 
what may be called the general trend or indication of the experi- 
ments, upon which it is founded, as the course of the present 
investigation seems to shew. 
Assuming that the radius function and its index m have been 
correctly ascertained, it may be eliminated from the values of log k’ 
by means of equation (33), m being taken either as 1°27, or as 
determined say by (38). In forming the values of log k',, Table A., 
the constant value for m has been assumed. As already remarked 
in §11, the quantities log &’ must be regarded as possibly functions 
of n, since the values of k” are so, from which they are derived ; 
an obvious fact when it is considered that each value of k” is 
determined by the intersection of the “n” lines—Figs. 2 to 6— 
with the axis of abscissw.1 Their arrangement according to the 
categories, in the order of the radii, or in the order of the values 
of n, fails to indicate any very definite relation. What indication 
there is of variation, is in favour of the assumption that &’ varies 
with », which after all is tantamount to a variation with the 
category. This indication may be noticed when the results of 
Table A. are plotted, or when the mean of series are taken, as in 
the table hereunder (K) and shewn in the illustrative plot, Fig. 9. 
The values k’,,(a) are calculated with m constant: (6) with it 
variable: only the (a) results are shewn in the figure. 
Table K. 
Nos, in Table. 12 47 5.9 10.6.8 11.13 17.12 19.15 14.20.18 
Mean v  B. 0-47 1°34 5-92 6-95 1-29 13°16 5-41 
1-70 1-77 178 181 184 187 1:91 1°97 
as log Ky, (a) 4-63 461 462 465 464 4:70 474 472- 
» yy log ki, (8) 474 4:80 4°61 470 4:70 482 475 476 
1 For when U=1 its logarithm is zero, and log k” is the 
to log I, gives the position of the axis of ordinates whore? I= L aan 
log I therefore zero, ' 
