FLUIDS AND THE DETERMINATION OF THE CRITERION. 125 
velocity and in exact accordance with the theoretical results obtained from the 
singular solution of the equation, when direct motion changes to sinuous, 7.¢., when 
DU, 
PES SS IS 
BL 
4. In the same paper I pointed out that the existence of this sudden change in the 
law of motion of fluids between solid surfaces when 
DU, = —K 
Pp 
proved the dependence of the manner of motion of the fluid on a relation between 
the product of the dimensions of the pipe multiplied by the velocity of the fluid and 
the product of the molecular dimensions multiplied by the molecular velocities which 
determine the value of 
- 
for the fluid, also that the equations of motion for viscous fluid contained evidence of 
this relation. 
These experimental results completely removed the discrepancy previously noticed, 
showing that, whatever may be the cause, in those cases in which the experimental 
results do not accord with those obtained by the singular solution of the equations, 
the actual motions of the water are different. But in this there is only a partial 
explanation, for there remains the mechanical or physical significance of the existence 
of the criterion to be explained. 
5. [My object in this paper is to show that the theoretical existence of an inferior 
limit to the criterion follows from the equations of motion as a consequence :— 
(1) Of a more rigorous examination and definition of the geometrical basis on 
which the analytical method of distinguishing between molar-motions and _heat- 
motions in the kinetic theory of matter is founded ; and 
(2) Of the application of the same method of analysis, thus definitely founded, to 
distinguish between mean-molar-motions and relative-molar-motions where, as in the 
case of steady-mean-flow along a pipe, the more rigorous definition of the geometrical 
basis shows the method to be strictly applicable, and in other cases where it is 
approximately applicable. 
The geometrical relation of the motions respectively indicated by the terms 
mean-molar-, or MzEAN-MerAn-Mortov, and relative-molar or RELAtTIVE-Mr&AN-Mortion 
being essentially the same as the relation of the respective motions indicated by the 
terms molar-, or MEAN-Morron, and relative-, or HEAT-Morion, as used in the theory 
of gases. 
I also show that the limit to the criterion obtained by this method of analysis and 
by integrating the equations of motion in space, appears as a geometrical linut to the 
