72 DESCRIPTION AND TRIAL OF THE 
Mr. Gatewood presented the paper, and in connection therewith, said :— 
“The subject of fluid friction has received attention in connection with shipbuilding, 
since 1869, as far as I have been able to find out, and in that year Mr. William Froude 
made public a theory of fluid friction in which he conceived that the velocity across the 
wake would vary uniformly. 
“In 1872, in the report of his first classical experiments on friction planes, he made a 
computation of the width of the wake, using the principle of momentum in the determina- 
tion, but still considering that the velocity across the wake would vary uniformly. 
“A few years ago Dr. Stanton published the results of some experiments he had made 
with air, in which he showed the curve of variation of velocity across pipes of various sizes. 
“Tt was this curve of Dr. Stanton’s that led me to take up the subject further and pre- 
pare this paper. A cross-section of the curve of velocity did not follow exactly the para- 
bolic law which it seemed would be theoretically correct. It rather indicated that the pres- 
sure across the section of the pipe was not uniform, but that the total head was uniform. A 
further consideration would show that this would be impracticable, and that somewhere in 
the length of a pipe, some distance from the entrance or exit, there must be a uniform pres- 
sure head in order to maintain constant conditions of flow. 
“The coefficients of viscosity, which have been published, and with which I have been 
able to get in touch, are of such a character that I have been unable to compare them with 
the coefficient which is used in the paper. They seem to have been determined, not with 
reference to the rate of change of velocity, but with reference to some other characteristic 
of the fluid. It would appear that if the theory here advanced is of value, this coefficient, 
based on the rate of change of velocity, should be experimentally determined. 
“This rate of change of velocity I have expressed in the paper as a trigonometrical 
function. I was advised that the proper way to describe it would be as the differential 
coefficient of the velocity; but it seemed to me that the general character of the paper would 
make it more satisfactory to treat it as a trigonometrical function than as a function of the 
calculus. 
“Formula (2) indicates that since tan varies as the ratio of the velocity to the linear 
dimension and also as the square of the velocity, if the velocity be doubled, the diameter 
of the pipe should be halved. That thought was first worked out on the basis that since ex- 
periments show that if the velocity were doubled, the diameter of the pipe must be halved 
in order to obtain the same cross-section of velocity, then the friction varied as the square 
of the velocity; in other words, that the formula proves backwards, if you take the experi- 
mental result as being correct. 
“Most of the publications on fluid friction state a critical velocity below which the fric- 
tion varies as the first power of the velocity, and above which it varies as some higher power ; 
but it would seem to me that if it is true, that if the velocity is doubled, the diameter of 
the pipe must be halved to obtain the same cross-section of velocity, then that fact proves 
that friction varies as the square of the velocity, the equivalent velocity of rubbing on the 
wall of the pipe. 
“By reference to Fig. 2, Plate 57, you will note that two curves are shown, one for a 
smooth pipe and the other for a rough pipe, and that while the coefficient of friction, ac- 
cording to the theory, in the case of the rough pipe would be 3.16 times as great as in the 
case of the smooth pipe, the resistance would be only 214 times as large. It would seem, 
then, if this is correct, that any comparison or discussion of friction based on the mean 
