STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 349 
19. The general equation for flow in circular pipes,—Summing 
up the results now reached, we may write for the mean velocity 
of the flow of water in a circular pipe under either régime, at 
any temperature, and with any radius, ‘slope,’ or material of pipe, 
D = |(#) "sna r]' (41) 
in which m depends upon the roughness of the channel, and can 
be set forth in categories, p and q are functions of the roughness 
expressed in m, and m is a function of the absolute dimensions of 
the pipe, sensibly, though perhaps not wholly independent of its 
roughness, but must be always taken as 2, whilen=1. The values 
of p,q and mare given in formule (39), (36) and (38) respectively. 
Reviewing the general result it seems evident :—(i.) That n is 
in some sense a measure of the intensity of the internal agitation 
developed by the rugosity of the boundary, or a measure of the 
integrated shear in a section. (ii.) That the decrease of the efficiency 
of fluidity in producing velocity parallel to the axis of the pipe, 
arises from the fact that the efficiency of the boundary condition in 
promoting internal agitation increases with fluidity :; this is prob- 
ably an asymptotic relation, for it seems certain that increase of 
fluidity continually promotes flow, though in less degree as the 
rugosity of the boundary increases: this view seems more obvious 
when a highly viscous liquid is studied : (iii.) That the variation 
in the index of the radius implies that the surface roughness is 
relative to the sectional area, throughout which it is the agent in 
promoting agitation. It may therefore be found, when sufficient 
exact experiments are to hand, that m is a function of both n 
and R, | 
The following are the mean values of » deduced in the preceding 
investigation, and values of g and q/n corresponding. 
The values in Tables IV. and V. are subject, the latter especially, 
to very great uncertainty as already pointed out ; and it has yet 
to be shewn by experiment that q, in the temperature function /%, 
is always positive, that is even when n is greater than 2. 
