Studies on the Motion of Viscous Flows — II 



refer to all of the hierarchies of uniformity and will have the same kind of gen- 

 eral relation to them as the known laws of classical mechanics have to the par- 

 ticular hierarchy of uniformity characterized by the equilibrium of forces. For 

 this purpose it is first necessary to identify and define descriptively the hier- 

 archies of uniformity in terms of forces, which as explained above, are inter- 

 preted here as the most fundamental, universal, and direct manifestation in ex- 

 perience, of the nonuniform connections existing between the universe and the 

 bodies contained within it. 



We may start by considering in some detail the very special and fundamen- 

 tal hierarchy of uniformity to which the known laws of classical mechanics per- 

 tain. This special hierarchy is defined by characteristics such that the vector 

 addition of all the nonuniform connections existing between a body and the uni- 

 verse which are posited in experience and which we designate by the name force, 

 sums to zero. It is clear that there can exist a conceivably infinite number of 

 distinct configurations of forces impressed on a material point, which individu- 

 ally designate the individual nonuniform connections between it and the universe, 

 and all of which equally belong to the very special hierarchy of force equilib- 

 rium. It is the differences between these distinct but otherwise equivalent force 

 configurations which I define as the hierarchies of uniformity. Figure 1 below 

 illustrates how we can conceive of an infinite number of distinct force configu- 

 rations, all of which belong to the hierarchy of uniformity defined by the equi- 

 librium of forces, and which by their differences here define the hierarchies of 

 uniformity. The figure shows various configurations of force equilibria, with 

 uniformity increasing from left to right. 



The hierarchies of uniformity, defined in terms of force fields, are now 

 used to formulate a Principle of Maximum Uniformity, which includes virtually 

 all the known laws of classical mechanics, as well as a general stability law. 

 The principle of maximum uniformity asserts that: among all the force config- 

 urations, individually characterized by force equilibrium, which can be collec- 

 tively and instantaneously accommodated in a finitely extended material domain 

 that is nonuniformly connected to the universe by maintained forces, the partic- 

 ular set of force configurations which actually evolves and which satisfies the 

 instantaneous and stringently exercized geometrical constraints, instantaneously 

 maximizes a global positive, definite, scalar measure of uniformity obtained by 

 summing the local measures of uniformity that depend on the local force con- 

 figurations over the entire domain. 



-A 



•^ 



^- 



Fig. 1 - Configurations of force of the 

 hierarchy of uniformity defined by the 

 equilibrium of forces 



471 



