FLUIDS AND THE DETERMINATION OF THE CRITERION. 129 
(b) That as the result of this definition the equations are true, and are only true 
as applied to fluid in which the mean-motions of the matter, excluding the heat- 
motions, are steady ; 
(c) That the evidence of the possible existence of such steady mean-motions, while 
at the same time the conversion of the energy of these mean-motions into heat is 
going on, proves the existence of some discriminative cause by which the periods in 
space and time of the mean-motion are prevented from approximating in magnitude 
to the corresponding periods of the heat-motions, and also proves the existence of 
some general action by which the energy of mean-motion is continually transformed 
into the energy of heat-motion without passing through any intermediate stage ; 
(d) That as applied to fluid in unsteady mean-motion (excluding the heat-motions), 
however steady the mean integral flow may be, the equations are approximately true 
in a degree which increases with the ratios of the magnitudes of the periods, in time 
and space, of the mean-motion to the magnitude of the corresponding periods of the 
heat-motions ; 
(e) That if the discriminative cause and the action of transformation are the result 
of general properties of matter, and not of properties which affect only the ultimate 
motions, there must exist evidence of similar actions as between the mean-mean- 
motion, in directions of mean flow, and the periodic mean-motions taken relative to 
the mean-mean-motion but excluding heat-motions. And that such evidence must be 
of a general and important kind, such as the unexplained laws of the resistance of 
fluid motions, the law of the universal dissipation of energy and the second law of 
thermodynamics ; 
(7) That the generality of the effects of the properties on which the action of trans- 
formation depends is proved by the fact that resistance, other than proportional to 
the velocity, is caused by the relative (eddying) mean-motion. 
(g) That the existence of the discriminative cause is directly proved by the 
existence of the criterion, the dependence of which on circumstances which limit the 
magnitudes of the periods of relative mean-motion, as compared with the heat-motion, 
also proves the generality of the effects of the properties on which it depends. 
(i) That the proof of the generality of the effects of the properties on which the 
discriminative cause, and the action of transformation depend, shows that—if in the 
equations of motion the mean-mean-motion is distinguished from the relative-mean- 
motion in the same way as the mean-motion is distinguished from the heat-motions— 
(1) the equations must contain expressions for the transformation of the energy of 
mean-mean-motion to energy of relative-mean-motion ; and (2) that the equations, 
when integrated over a complete system, must show that the possibility of relative- 
mean-motion depends on the ratio of the possible magnitudes of the periods of relative- 
mean-motion, as compared with the corresponding magnitude of the periods of the 
heat-motions. 
(7) That when the equations are transformed so as to distinguish between the 
MDCCCXCY,—A. s 
