STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS, 321 
and Reynolds clearly recognise that as the so-called critical velocity 
is approached the condition of flow becomes unstable. Hagen is 
not committed to any definitive doctrine as to the limits of stability 
of the rectilinear régime, but Reynolds has given an expression 
for the velocity, beyond which that régime cannot, he supposes, 
be maintained. His expression has however, certainly not been 
tested between sufliciently wide limits to justify the conclusion he 
makes from his experiments. For centimetre units, Reynold’s 
formula may be written 
ye UE eas ..(6) 
U, being the alleged critical velocity, / the relative fluidity, (1/P 
in Reynold’s formula) and-& the radius of the pipe. The numerical 
cotflicient is for glass pipes, and great steadiness. In Darey’s 
experiments the highest velocity of the rectilinear régime is only 
about one-sixth of the above amount, when it passes into the 
so-called second régime, sv that the coéfficient in (6) would require 
to be reduced to about 18 to apply to Darcy’s pipes. 
Reynold’s conclusion as to the existence of a definite critical 
velocity, discoverable in his formula, is questionable. Lord Kelvin 
in papers on “The steady motion of fluids,” says :—‘‘ It seems 
“probable, indeed almost certain, that analysis similar to § 38 and 
“S$ 39,” of his papers, “will demonstrate that steady motion is. 
stable for any viscosity however small,” and that practical 
unsteadiness is to be “explained by the limits of stability becoming 
narrower and narrower the smaller the viscosity.” Although the 
force of these observations was hardly admitted by Lord Rayleigh,’ 
they are strongly endorsed by Rudski.‘ Perhaps the best way of 
* Phil. Mag. 1887 o pp. 459-464; 529-539; 1887 (2) pp. 188-196; 
272-278; 34 2 
2 The ‘iiss are mine. 
Rayleigh’s papers on the “pees are :—(a) ‘On the Instability 
of certain fluid motions.”—Proc. Lond. Math. Soc. 7 Bop ohe 1880. (6) 
“On e Question of the stability of the Flow of Liqui — Phil. Mag. 
1892, Vol XXXIV., pp. 59-70; 145-154; 177-180. 
ey ey on the flow of water in a straight pipe.” —Phil. mig: 1893 a) 
439 - 
U—Dee. 1, 1897, 
