322 G. H. KNIBBS. 
satisfying oneself as to this point, is to observe the circumstances 
of flow when they are such that one may establish either régime 
at will. The ease with which the rectilinear régime may be 
disestablished increases with increase of velocity and with increase 
of fluidity. ‘The variableness of the relations between velocity 
and ‘hydraulic gradient’ (/) in the region lying between what 
may be called—speaking relatively—the stable linear and stable 
turbulent régimes—admirably illustrated in Hagen’s curves of 
velocity, and by Reynold’s experiments also—is a further indica- 
tion that the view of Sir Wm. Thomson is correct. 
6. The rationalization of Reynold’s formula for rectilinear flow 
in pipes.—Reynolds has given in his paper, before referred to, an 
alleged general formula for flow in pipes, which may be written 
OG oe gee SS 2 a8 a (7) 
M and N being constants for all classes of pipe, f the relative 
fluidity, and R and U having the same meanings as before.’ This 
he says, holds for every pipe and every condition of water: 4 
statement which demands further examination. Considering for 
the present only the case for rectilinear flow, the exponent ” is 
then unity ; — since H/L = J, the above expression may be 
written 
U= = PTR IB) 
‘Comparing this with (1) and remembering that 1/y=//7., we Se@ 
that the ratio of Reynold’s factors is simply 
a gp 
2a = 8 1, ee ee (9) 
If these expressions were identically equal, the rationalization 
would be complete; but they are not so: the factors Mand ¥ 
are merely empirical. Thus while (1) is a rational, (7) and 0 
are only empirical formule. 
Proceeding to the testing of Reynold’s ratio, it may be — 
that for engineering purposes the variations of density (p) and of 
gravity (y) may be generally neglected, and consequently U/N 
1M = 8A,and N = 2B, and/ = 1/P in Reynolds’s formula. 
