130 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
mean-mean-motions, of infinite periods, and the relative-mean-motions of finite periods, 
there result two distinct systems of equations, one system for mean-mean-motion, as 
affected by relative-mean-motion and heat-motion, the other system for relative-mean- 
motion as affected by mean-mean-motion and heat-motions. 
(7) That the equation of energy of mean-mean-motion, as obtained from the first 
system, shows that the rate of increase of energy is diminished by conversion into 
heat, and by transformation of energy of mean-mean-motion in consequence of the 
relative-mean-motion, which transformation is expressed by a function identical in 
form with that which expresses the conversion into heat; and that the equation of 
energy of relative-mean-motion, obtained from the second system, shows that this 
energy is increased only by transformation of energy from mean-mean-motion 
expressed by the same function, and diminished only by the conversion of energy 
of relative-mean-motion into heat. 
(x) That the difference of the two rates (1) transformation of energy of mean-mean- 
motion into energy of relative-mean-motion as expressed by the transformation 
function, (2) the conversion of energy of relative-mean-motion into heat, as expressed 
by the function expressing dissipation of the energy of relative-mean-motion, affords 
a discriminating equation as to the conditions under which relative-mean-motion 
can be maintained. 
(/) That this discriminating equation is independent of the energy of relative-mean- 
motion, and expresses a relation between variations of mean-mean-motion of the first 
order, the space periods of relative-mean-motion and jz/p such that any circumstances 
which determine the maximum periods of the relative-mean-motion determine the 
conditions of mean-mean-motion under which relative mean-motion will be maintained 
—determine the criterion. 
(m) That as applied to water in steady mean flow between parallel plane surfaces, 
the boundary conditions and the equation of continuity impose limits to the maximum 
space periods of relative-mean-motion such that the discriminating equation affords 
definite proof that when an indefinitely small sinuous or relative disturbance exists 
it must fade away if 
P D U, |p 
is less than a certain number, which depends on the shape of the section of the 
boundaries, and is constant as long as there is geometrical similarity. While for 
greater values of this function, in so far as the discriminating equation shows, the 
energy of sinuous motion may increase until it reaches to a definite limit, and rules 
the resistance. 
(n) That besides thus affording a mechanical explanation of the existence of the 
criterion K, the discriminating equation shows the purely geometrical circumstances 
on which the value of K depends, and although these circumstances must satisfy 
geometrical conditions required for steady mean-motion other than those imposed by 
