44 Theory of Incompressible Viscous Fluids, fyc. [May 24, 



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 transform- 

 ation of energy from mean-mean-motion expressed by the same 

 function, and diminished only by the conversion of energy of relative- 

 inean-motion into heat. 



(&.) 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 func- 

 tion 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. 



(Z.) 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 ftjp 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 con- 

 tinuity 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 



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 geo- 

 metrical 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. 



(.) 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 the 

 conservations of mean energy and momentum, the theory admits of 

 the determination of an inferior limit to the value of K under any 

 definite boundary conditions, which, as determined for the particular 

 case, is 



517. 



