FLUIDS AND THE DETERMINATION OF THE CRITERION. 131 
the conservations of mean energy and momentum, the theory admits of the determi- 
nation of an inferior limit to the value of K under any definite boundary conditions, 
which, as determined for the particular case, is 
o17. 
This is below the experimental value for round pipes, and is about half what might 
be expected to be the experimental value for a flat pipe, which leaves a margin to meet 
the other kinematical conditions for steady mean-mean-motion. 
(0) That the discriminating equation also affords a definite expression for the 
resistance, which proves that, with smooth fixed boundaries, the conditions of 
dynamical similarity under any geometrical similar circumstances depend only on the 
value of 
dp 5. 
z Fis b%, 
where 0 is one of the lateral dimensions of the pipe ; and that the expression for this 
resistance is complex, but shows that above the critical velocity the relative-mean- 
motion is limited, and that the resistances increase as a power of the velocity higher 
than the first. 
Section II. 
The Mean-motion and Heat-motions as distinguished by Periods.—Mean-mean- 
motion and Relative-mean-motion.—Discriminative Cause and Action of Trans- 
formation.—Two Systems of Equations.—A Discriminating Equation. 
6. Taking the general equations of motion for incompressible fluid, subject to no 
external forces to be expressed by 
du 
a, d A 
pa =T le (Pic + pu) + ay (Pye + puv) + = (Pie + puw) | | 
dy d Gp d 
Pig =~ [ap (Ba + pm) + 5 (Par + p20) + 5 (Py + pow) b & . (1), 
a | 
dz J 
U L 1 
Pp a ania ee (Diz -F pwi) ae a (Py: oF pwr) ay 
dz 
(Pes + pwr) 
with the equation of continuity 
O=du/dx + dojdy--dwdz. . . . . . +. (2); 
where p,,, &c., are arbitrary expressions for the component forces per unit of area, 
resulting from the stresses, acting on the negative faces of planes perpendicular to 
$2 
