STEADY FLOW OF WATER IN UNIFORM PIPES AND CHANNELS. 327 
a series of fictitious particles the movements of which, parallel to 
that axis are the means of the similar movements of particles in 
the same relative position, these fictitious particles will lie also 
on a conoidal surface but not that of a paraboloid, the conoid 
being much flatter at the apex, see Fig. 1, in which the points at 
one-third and two-thirds of the diameter are plotted from the 
mean of Darcy’s experiments. 
The problem of turbulent flow was attacked by Boussinesq in 
1872, in his incomparable “Essai sur la théorie des eaux covrantes.”” 
His method of analysis is as follows :—The real velocities are con- 
sidered to rapidly and abruptly change from point to point in any 
section, and thus to produce a degree of friction of quite another 
and greater order of magnitude than can occur in the rectilinear 
régime. The mean action across any fixed plane element is 
measured, not merely by the mean local velocities or by their first 
derivatives defining the rate of shear of the fluid, but also by the 
intensity of agitation at the point considered. The causes of 
the agitations having been ascertained, the coéfficient of internal 
friction is made to vary with them. As equations of motion are 
selected, not those which express, at a given instant, the dynamic 
equilibrium of different elementary volumes of the fluid, but the 
mean of these during a short but sufficient time ; so that one is 
able to call them, the equations of the mean dynamic equilibrium 
of the fluid particles which successively pass any particular point.” 
This statement of part of the great problem, to the solution of 
which Boussinesq applied himself, presents a definite conception 
of the nature of the movement. The analysis of the problem now 
attempted has for its object, the discovery, without reference to 
-Boussinesq’s deductions, of the mathematical form in which the 
results of observation can be consistently expressed, so that they 
will really represent the observed phzenomena. 
1 Mém. des Savants étrangers, t. 23, pp. 1 - 680, 1877. 
2 Loe. cit., pp. 6, 7. 
