1894.] Theory of Incompressible Viscous Fluids, $-c. 43 



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 cavise 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 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 ; 



(/.) That the generality of the effects of the properties on which 

 the action of transformation depends is proved by the evidence that 

 resistance, other than proportional to the velocity, is caused by the 

 relative (eddying) mean-motion. 



(</.) That the existence of the discriminative cause is directly proved 

 the existence of the criterion, the dependence of which on circum- 

 stances 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. 



(&.) That the proof of the generality of the effects of the proper- 

 ties on which the discriminative cause and the action of transforma- 

 tion depend, shows that if in the equations of motion the ruean-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 ; 

 ind (2) that the equation, when integrated over a complete system,, 

 lust show that the possibility of relative-mean-motion depends on 

 the ratio of the possible magnitudes of the periods of relative-mean- 

 motiou, as compared with the corresponding magnitude of the periods 

 the heat-motions. 



(i.) That when the equations are transformed so as to distinguish 

 jtween the mean-mean-motions of infinite periods and the relative- 

 ican-motion of finite periods, there result two distinct systems of 

 (nations, one system for mean-mean-motion, as affected by relative- 

 lean-motions and heat-motion, the other system for relative-meau- 

 lotion as affected by mean-mean-motion and heat-motions. 



(_/.) That the equation of energy of mean-mean-motion, as ob- 

 tained 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- 



