IV. On the Dynamical Theory of Incompressible Viscous Fluids and the 
Determination of the Criterion. 
By OssorneE Reynotps, V.A., LL.D., F.RS., Professor of Engineering in Owens 
College, Manchester. 
Received April 25—Read May 24, 1894. 
Section I. 
Introduction. 
1. THE equations of motion of viscous fluid (obtained by grafting on certain terms to 
the abstract equations of the Eulerian form so as to adapt these equations to the case 
of fluids subject to stresses depending in some hypothetical manner on the rates of 
distortion, which equations NAVIER* seems to have first introduced in 1822, and 
which were much studied by Cavcuyt and Porsson{) were finally shown by 
St. Venant§ and Sir Gasrret SToKEs,|| in 1845, to involve no other assumption than 
that the stresses, other than that of pressure uniform in all directions, are linear 
functions of the rates of distortion, with a co-etlicient depending on the physical state 
of the fluid. 
By obtaining a singular solution of these equations as applied to the case of 
pendulums in steady periodic motion, Sir G. SrokeEs was able to compare the 
theoretical results with the numerous experiments that had been recorded, with the 
result that the theoretical calculations agreed so closely with the experimental 
determinations as seemingly to prove the truth of the assumption involved. This 
was also the result of comparing the flow of water through uniform tubes with the 
flow calculated from a singular solution of the equations so long as the tubes were 
small and the velocities slow. On the other hand, these results, both theoretical and 
practical, were directly at variance with common experience as to the resistance 
* ‘Mém, de Académie,’ vol. 6, p. 389. 
+ ‘Mém. des Savants Ktvangers,’ vol. 1, p. 40. 
t ‘Mém. de l’Académie,’ vol. 10, p. 345, 
§ «B.A. Report,’ 184.6. 
|| ‘Cambridge Phil. Trans.,’ 1845. 
9 ‘Cambridge Phil. Trans.,’ vol, 9, 1857. 
R 2 6.5.95 
