140 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
Expressions for the Rate of Transformation and the Discriminative Cause. 
14. It has already been shown (Art. 8) that the equations of motion approximate 
to a true expression of the relations between the mean-motions and stresses, when the 
ratio of the periods of mean-motions to the periods of the heat-motions approximates 
to infinity. Hence it follows that these equations must of necessity include whatever 
mechanical or kinematical principles are involved in the transformation of energy of 
mean-mean-motion to energy of relative-mean-motion. It has also been shown that 
the properties of matter on which depends the transformation of energy of varying 
mean-motion to relative-motion are common to the relative-mean-motion as well as to 
the heat-motion. Hence, if the equations of motion are applied to a condition in 
which the mean-motion consists of two components, the one component being a mean- 
mean-motion, as obtained by integrating the mean-motion over spaces §, taken about 
the point x, y, z, as centre of gravity, and the other component being a relative-mean- 
motion, of which the mean components of momentum taken over the space S, every- 
where vanish, it follows :— 
(1) That the resulting equations of motion must contain an expression for the rate 
of transformation from energy of mean-mean-motion to energy of relative-mean- 
motion, as well as the expressions for the transformation of the respective energies of 
mean- and relative-mean-motion to energy of heat-motion ; 
(2) That, when integrated over a complete system these equations must show that the 
possibility of the maintenance of the energy of relative-mean-motion depends, whatsoever 
may be the conditions, on the possible order of magnitudes of the periods of the relative- 
mean-motion, as compared with the periods of the heat-motions. 
The Equations of Mean- and Relative Mean-Motion. 
15. These last conclusions, besides bringing the general results of the previous 
argument to the test point, suggest the manner of adaptation of the equations 
of motion, by which the test may be applied. 
Put 
“U=u+y, v=v+v, eee 4 5 5 (iil), 
where 
Li OP(HM PH) etonicss ees 6 6 6 6 6 5 (2), 
the summation extending over the space S, of which the centre of gravity is at the 
