212 
MESSRS. T. E. STANTON AND J. R. PANNELL ON SIMILARITY OF 
Unfortunately in some of these observations the temperature has not been recorded 
so that the values of vd/v calculated are only approximate. The reduced results 
have been plotted in fig. 5 where, for convenience of comparison, the limits of the 
results of the present experiments have been indicated by broken lines. In the case 
of Darcy’s results the points referring to the two lead pipes and the bitumen-covered 
pipes of 8'6 and 19'6 cm. diameter have been taken as forming one group with 
approximately the same surface smoothness and a line has been drawn through their 
mean position. This line shows the same characteristics as the curve of the present 
experiments, except that the surface is somewhat rougher. The bitumen-covered pipe 
of 28'5 cm. diameter had evidently a considerably rougher surface, but otherwise the 
distribution of the points are what would be expected. For Reynolds’s experiments 
on the two lead pipes a mean line has also been drawn which indicates a lower resist¬ 
ance than that found in the pipes used for the present experiments, but the character 
of the curve and the position of the critical point are identical with those now found. 
From the point of view of comparison with the present observations neither of the 
above sets of experiments are so satisfactory as those of Saph and Schoder, # since the 
nature of the surfaces in the two researches was probably identical. The plottings of 
these, with a very few exceptions, lie within the limits of the present experiments of 
which they form an excellent check. The whole series of these experiments include 
observations on pipes of 5*32, 3*81, 3'14, 2'68, 2'08, 1’60,1'27, 0'95, 0*82, 072, 0'66, 0'56, 
0'46, 0*38 and 0'272 diameter, of which only the 5'32, 3*81, 3'14, 2*68, 1'60, 0'82 and 
0*272 cm. have been reduced for plotting in fig. 5. For the determination of the fric¬ 
tional resistance in the case of the flow of air the amount of reliable data appears to be 
very small. The six experiments of Dr. BrixI" on iron pipes of 8*26, 13*2, and 17*1 cm. 
diameter are in fair agreement with the present results, but the interpretation of the 
series of observations on the 0*635 cm. lead pipe is difficult as the form of the curve 
appears to indicate that the conditions are those of stream-line flow. The only expla¬ 
nation which can be suggested is that the diameter of the pipe was over-estimated by 
some 15 per cent., in which case the curve shown through the plotted points would 
come into fair agreement with the theoretical curve for stream-line flow. The results 
of Stockalper’s experiments, which possess additional interest owing to the fact that 
the air was under a pressure of about 5 atmospheres, are well in agreement with the 
present experiments as regards the 20 cm. pipe and less so in the case of the 15 cm. 
pipe, but unfortunately the range of velocity is not a large one. 
In conclusion, a note on the various deductions from theoretical considerations of 
the criterion for similarity of motion in fluids may be of interest. In a paper read 
before the Cambridge Philosophical Society in 1850, Stokes^ considered any number of 
similar systems composed of similar solids, oscillating in a similar manner in different 
* ‘ Proc. Amer. Soc. Civil Engs.,’ 1903, vol. 51, p. 253. 
t ‘Phil. Mag.,’ vol. 17, 1909, p. 395. 
| Stokes, ‘ Mathematical and Physical Papers,’ vol. 3, p. 17. 
