STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 3395 
at which the velocity would be unity, 1 cm.—and it defines the 
point at which the m lines Fig. 2 — 6 intersect the axes of abscisse. 
An examination of the values of the coéfficient kh’, shews, that 
not only is it a function of the radius of the pipe, but also of its 
rugosity. It may also be assumed to be a function of the temper- 
ature, and therefore of the fluidity of the water. Hence, if the 
index n be treated as itself a function of the roughness of the pipe, 
or rather of the degree of vortex agitation which is set up in the 
fluid by the agency of the boundary conditions, then we may put 
as certainly true, k” = (/ R), and as probably true k” = ¢( fin.R). 
12. Experimental proof that the temperature function is f', 
SJ being the ‘flwidity.’—In his 1853 experiments previously referred 
to, Hagen obtained a formula expressing the flow in his pipes, 
which may be written 
U* =o RL (22) 
in which m was on the average about 1-774, but was taken as 1-75, 
and mas 1:25. The values of the logarithms c’—the reciprocals 
of Hagen’s (m) quantities—for the temperatures given by him, 
expressed in Celsius instead of Reaumur degrees, are as follows :— 
Table 
Temperature 0. 183° 31}° 433° 62° 81}° 
Values oflog f -234 359 463 591 698 
Value of loge’ 4:4854 4:5185 4:5414 4:5756 4:5985 
The values of log f are taken from Tables I. and II.; and for the 
higher temperatures are practically a mean of Slotte’s (1893) and _ 
Thorpe and Rodger’s (1894) corrected values. 
From the above equation it is obvious that the YU" « ec’ when 
&"T is constant: consequently c’ is a function of the fluidity. 
Now since the increase of fluidity of any liquid flowing through a — 
tube, facilitates the intensity of the internal agitation, very much 
in the same manner as increase of velocity would intensify it, 
when once the stability of the rectilinear régime has been over- 
come, it may be supposed likely that the relation 
Cm OF * iiiecind(ae) 
