150 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
The existence of a criterion is thus seen to depend on the existence of certain 
restrictions to the value of the periods of relative-mean-motion—on the existence of 
conditions which impose superior limits on the values of a. 
Such limits to the maximum values of a may arise from various causes. If du/dy 
is periodic, the period would impose such a limit, but the only restrictions which it is 
my purpose to consider in this paper, are those which arise from the solid surfaces 
between which the fluid flows. These restrictions are of two kinds—restrictions to 
the motions normal to the surfaces, and restrictions tangential to the surfaces—the 
former are easily defined, the latter depend for their definition on the evidence to be 
obtained from experiments such as those of PorsEUILLE, and I shall proceed to show 
that these restrictions impose a limit to the value of a, which is proportional to D, 
the dimension between the surfaces. In which case, if 
J (la/dyy? = U/D, 
equation (26) affords a proof of the existence of a criterion 
DLO Cee Gt aera yg (AT) 
of the conditions of mean-mean-motion under which relative or sinuous-motion can 
continuously exist in the case of a viscous fluid between two continuous surfaces 
perpendicular to the direction y, one of which is maintained at rest, and the other in 
uniform tangential-motion in the direction # with velocity U. 
Section III. 
The Criterion of the Conditions under which Relative-mean-motion cannot be main- 
tained in the case of Incompressible Fluid im Uniform Symmetrical Mean-flow 
between Parallel Solid Surfaces.—Expression for the Resistance. 
25. The only conditions under which definite experimental evidence as to the value 
of the criterion has as yet been obtained are those of steady flow through a straight 
round tube of uniform bore; and for this reason it would seem desirable to choose 
for theoretical application the case of a round tube. But inasmuch as the application 
of the theory is only carried to the point of affording a proof of the existence of an 
inferior limit to the value of the criterion which shall be greater than a certain 
quantity determined by the density and viscosity of the fluid and the conditions of 
flow, and as the necessary expressions for the round tube are much more complex 
than those for parallel plane surfaces, the conditions here considered are those defined 
by such surfaces. 
