STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 343 
not large, is adverse to Reynolds’ theory, which therefore may be ; 
rejected 
The means of the experimental results, in which alone the real 
relation may be supposed to be disclosed in the presence of 
apparently hopeless anomalies in individual cases, indicates an 
average relation, k’ « R!, as generally interpreting the experi- 
ments within their own limits; and this in the mean with a 
precision that could hardly have been anticipated. That this 
index is well determined cannot of course be alleged : experiments 
will have to be made with a far higher order of precision than in 
the past, before either the general constancy of the index can be 
really assured, or its exact value or law of variation accurately 
ascertained. 
From some considerations it might seem probable that m would 
vary with F itself (case iii,), and indeed also with n, the latter 
imply ing perhaps, that the roughness must be considered in rela- 
tion to the dimensions of the pipe; a view from which there seems 
to be no escape, if extreme cases be contemplated. This view how- 
ever is not supported directly by the order of the variations in 
the values of n—that is with the measure of roughness—shewn in 
Table A., excepting perhaps in line 17; but here the individual 
results are so inconsistent that no reliance can be placed upon the 
value of n derived therefrom. As however, there is some evidence 
that m is not constant, this question will be further considered, 
viz., in the section next following, § 17. 
It is not unimportant to remember that a limited number of 
experiments may, through errors of observation, often suggest a 
relation, which a larger series will shew to be quite accidental. 
For example in considering the value of m, it might appear from 
Darcy’s experiments with lead pipes, that m increases with R, see 
Table H. 3 - 4, 3-5, and 4-5; or from those with iron pipes, 
that it diminishes with R, see 7-9, 8 - 10; and 11 - 13, 12 - 13; 
and soon. This may possibly be accounted for by the fact that 
m isa function both of n and R, which at any rate must be_ 
admitted if a general formula is to be reached. No matter how _ 
