206 
MESSRS. T. E. STANTON AND J. R. PANNELL ON SIMILARITY OF 
the third is not in agreement. It is possible that this may be due to the third pipe 
being rougher in surface than the others, as it has been shown by direct observation 
of distribution of velocity in rough and smooth pipes at the National Physical 
Laboratory* that the ratio of mean speed to maximum speed in a rough pipe is less 
than that in a smooth one. 
The interpretation of Darcy’s results is also difficult for the same reason. 
According to the determinations of the frictional resistances of these cast-iron pipes 
given by Darcy, their surfaces are all appreciably rougher than those used in the present 
experiments, so that the ratio in the case of the former should be less than for the 
latter, but no direct evidence of this is obtained from the plotted results. In fig. 1 
is also plotted the ratio obtained by direct integration from the radial distribution of 
the velocity in a 5'08 cm. pipe, which agrees well with the other results. 
The Surface Friction Experiments. 
The ratio of the mean velocity to the velocity at the axis having been determined, 
it was possible to complete the observations of surface friction for those cases in which 
no other method of estimating the discharge was available. 
The scope of the experiments will be seen from the table on p. 207 in which the 
dimensions of the pipes, and the methods of setting up the flow and estimating the 
velocity and friction per unit area are stated. The actual values of the mean velocity 
of flow in centimetres per second and the surface friction in dynes per square 
centimetre of the surface are given in the tables at the end of the paper together 
with the calculated values of the ratio vd/v (Tables II. and III.). 
The method of representing the results is that, suggested by Lord Rayleigh, of 
plotting points whose ordinates are the values of R /pv 2 and abscissae the corresponding 
values of vdfv, with the previously stated modification that the logarithms of vdjv are 
taken as the abscissae instead of the actual values. In this way it has been found 
possible to include all the results on a reasonable length of diagram, aud at the same 
time to show the region of the change from eddying to stream-line motion on a 
fairly open scale. This has been done in fig. 3. 
The extension of the experiments for values of vdjv greater than 115,000, which 
was the maximum attainable with the appliances available at the National Physical 
Laboratory, has been made possible by the kindness of Mr. Edward B. Ellington, 
Engineer of the London Hydraulic Power Supply Company, who has allowed the 
authors to connect their experimental pipes to the high-pressure water main 
at the Grosvenor Road Pumping Station. In this w r ay it lias been found possible to 
reach velocities of 3200 cm. per second in the 1'255 cm. diameter pipe, 5250 cm. per 
second in the 07125 cm. pipe, and 5600 cm. per second in the 0'361 cm. pipe. The 
maximum value of vd/v attained was 430,000. 
* ‘Roy. Soc. Proc.,’ A, vol. 85, p. 371. 
