88 BULLETIN 376, U. S. DEPARTMENT OF AGRICULTURE. 
Gardner S. Williams, formerly of Cornell and Michigan Universities, as representing 
the best average value. They inclined rather toward varying the exponent of V within 
the range of about 1.75 to 1.95 and they also varied the coefficient m for different kinds 
and conditions of pipes from about 0.30 to 0.50. 
The exponent 1.25 was dete rmin ed by Schoder from a plotting of all known observa- 
tions on pipes covering the entire field from very small drawn-brass pipes to very 
large tuberculated riveted-steel pipes, and is therefore not based on estimate nor 
opinion, but on actual facts. ' The writer's observations on wood-stave pipes developed 
a value of 1.26, which he has since reduced in the formula for practice to 1.25 to con- 
form to the generally accepted figure. 
In the matter of proposing a new formula, the writer lives in a glass house and can 
therefore not afford to throw stones. However, there were extenuating circumstances 
in this case, which need not be discussed here, that ultimately led to a rather wide 
acceptance of the writer's formula, although he originally offered it with reservations 
because there were a number of points not satisfactorily explained, and experiments 
on very large pipes were notably lacking. Thanks to the author's careful experiments 
the deficiency has now been supplied to a considerable extent and we are closer to the 
ultimate solution. 
It is interesting to note that the author has arrived at the same figure for exponent of 
V, namely 1.8, that developed from the writer's experiments. Unfortunately, he 
had not time to check the author's analysis on this point, nor to make an independent 
analysis of the data in this paper, but the evidence in support of this average figure 
may now be fairly considered as having been greatly extended. The evidence in 
support of 1.25 as the exponent of d is in the writer's opinion equally as good if not 
better, since its derivation was based on observations on all kinds and sizesof pipes. 1 
If we could, therefore, now agree on these two figures and throw the variation in 
formula for different classes of pipe into the factor m, a much longer step will have been 
made toward the general acceptance by engineers and courts of an exponential formula 
for flow of water in pipes. 
It must be conceded that additional experiments that may be made in the future, 
especially in the field of diameters between 60 and 160, may have a marked effect on 
the exponent of d and we will then no doubt be confronted by another formula. More- 
over, as has already been pointed out, the personal equation of the analyst, especially 
in the application of weights and methods of reasoning, must be taken into account 
and another person with the same data as a basis would no doubt arrive at somewhat 
different conclusions than those given in the present paper. No doubt this factor has 
been eliminated to the greatest possible degree in the present paper, but complete 
i Author's footnote (the italics are his).— A study of original sources does not bear out this statement. In 
Trans. Amer. Soc. Civ. Engin., vol. 51, Messrs. Saph and Schoder, on page 305, state: "First of all, the line 
best fitting the points for the writers' brass pipes has been drawn." which line, as they say on page 306, 
" forms the lower limit of the zone in which all the plotted points lie. This is another way of stating that 
these pipes represent the extreme of smoothness and ideal conditions." The exponent of d for this line 
was found to be 1.25. (This line is A, fig. 7.) The values of m for all kinds of pipes were then platted and 
Saph and Schoder state (id., p. 308): "It will be seen that a line parallel to the one already drawn will 
represent fairly the other limit ' ' of the zone. (This line is shown as B, fig. 7. ) Thus it will be seen that this 
exponent was derived from a study of smooth brass pipes and then applied to all kinds of pipes by drawing a 
line parallel, i. e., with the same slope, hence indicating the same exponent of d, or 1.25. In discussing the 
exponent as derived by Saph and Schoder, Mr. Allen Hazen states (id., p. 321), "It is, unfortunately, a 
fact that probably none of the large pipe has as smooth surfaces as the brass pipe used by the authors (Saph 
and Schoder) and some allowances must be made for this fact in the comparisons, and, while the line drawn 
as a general average would be a matter of individual judgment, the writer (Hazen) would hardly give it 
aw inclination greater than that corresponding to an exponent of 1.20. If the matter were carried further the 
exponent would probably be lower." It is interesting to again note that the exponent of d is 1.17 in the 
formula derived from a study of these same pipes of all kinds by Williams and Hazen, and that this is the 
same exponent found in this study of the flow in wood-stave pipes. (Line E, fig. 7. This question is 
further discussed on page 94.) 
