STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 323 
treated as a constant, which strictly of course it is not, being a 
function of gravity and of the density of the column of fluid by 
means of which the difference of pressure at two points in the 
supposed-horizontal pipe is measured. Accepting the results given 
in Table I., taking the value of g for latitude 45° and sea-level, 
as 980-61, we have, for water, 
> = 6851:26 or 6145°6 at 15°C.......... (10) 
the density being regarded as -999173 at the latter tempera- 
ture. Reynolds gives for 4 M the value 67°70 and for $ V 0-03963, 
both reduced for the centimetre unit. Hence according to him, 
the above ratio is 6833-2 a value which, though sensibly the same 
as that above (10), is very probably slightly under the truth. The 
difference, 0-18%, is quite negligible from an engineering point of 
view. The logarithm of 6145-6 is 3-8354: obviously the compu- 
tation of (1) or (8) logarithmically, leaves nothing to be desired 
on the score of simplicity. 
In the rationalized form, the formula for rectilinear flow in 
circular pipes is not Reynold’s, but Neumann’s: it was given some 
time prior to 1860. Had Reynold’s expression really held for 
both régimes, then although empirical, its generality would have 
commended it. The question of its general correctness will be 
hereinafter examined. 
7. Hagens’s discussion of linear and non-linear flow in pipes, 
in 1858.—As already remarked, Hagen, as far back as 1853, 
observed that fluids were subject not only toa rectilinear, but 
also to a turbulent or eddying condition of flow ; and that in the 
latter case some of the energy was ineffective in the production of 
motion parallel to the axis of the pipe, since it was expended in 
the eddying agitation. The tabulated results and diagrams by 
means of which his treatise was illustrated, shewed very clearly 
the significance of the change from the rectilinear to the non- 
rectilinear régime. Hagen’s experiments were made mainly with 
three tubes of sheet brass soldered into the form of pipes, their radii 
in centimetres being A “14083, B -20242 and C +2974 at 15° B, 
