RECENT PROGtRESS IN HYDRODYNAMICS. 79 



coefficient in a mixture of gases. Later experiments ' have shown that 

 the proportionality of the viscosity to absolute temperature docs not hold 

 for all gases. But Maxwell's work shows in any case that the connection 

 between them depends on the law of force, which is certainly a complicated 

 one, and not likely a priori to be the inverse fifth. Possibly for an 

 attractive force the viscosity might decrease with increase of temperature, 

 and if so would form an answer to Stefan's objection against making the 

 same explanation of friction a basis for a theory of the viscosity of liquids. 

 Lamb, in his treatise on hydrodynamics, deduces the equation by a method 

 based on those of Stokes and St. Venant. Boussinesq ^ applies the 

 equations for an elastic solid directly. 



M. Levy ^ attempts to show that the stress cannot be represented by 

 functions of the form : — 



but that the exact expressions are of the form — 



dv . dw" 



' c. 



X.= (.H-...v-)('£+|)... 



whilst M. Kleitz,'* arriving at the same result as to the error of the ordinary 

 method of expressing the stresses, starts by supposing the constants to vary 

 for different small planes round a point, and tries to find the law of this 

 variation. Meyer ^ has also deduced equations of motion on the supposi- 

 tion that the action between two particles takes time to act. 



In the ' Proceedings ' of the London Mathematical Society, Butcher ^ 

 has attempted to develop a general theory which would comprise the 

 theory of elastic solids and perfect fluids as particular cases. He supposes 

 a body composed of small groups of molecules differing from one another, 

 which he divides into two classes : (A) which recover their original form 

 after being subjected to a strain less than a certain amount, and (B) those 

 in which this limiting strain is zero. When only A groups are present we 

 have an elastic solid ; when A and B groups are present the body partially 

 returns to its original configuration after a strain, but the amount of its 

 return is a function of the previous duration of the strain ; if the body 

 does not return at all, B groups only are present, and we get a viscous 

 or perfect fluid according as the acting stress is finite or infinitesimally 

 small. With varying proportions of A and B groups appear difierent 



' For a discussion of this point tlie reader is referred to O. E. Meyer's Die Kine- 

 tisclie Thforic dcr Gaitc. Breslan, 1S77, p. 157. Also for a fidl abstract of what is 

 known experimentally on the subject. 



2 ' Memoire sur Tinfluence des frottements dans les mouvements reguliers des 

 fluides.' LumvUle (2) xiii. p. 377. 



' ' Rapport sur un ]\I6mou-e de M. Mam-ice Levy relatif a, rhydrodynamiqne des 

 liquides homog&nes, particulii^rement il leur econlement rectilig-ne et permanente.' 

 Comjrtes Rendus, Ixviii. p. 582. 



■• ' Rapport, sur un Memoire de M. Kleitz, intitule etudes sur les forces moleculaires 

 dans les liquides en mouvement, et application ii I'hydrodynamique.' Comptes 

 Rendus, Ixxiv. p. 426. 



» ' Zur Theorie der inneren Reibuug,' Borcli. Ixsviii. p. 130. Zusatz zu der 

 Abhandlung ' Zur Theorie der inneren Reibung,' Borch. Isxx. p. 31, "5. 



« ' On viscous fluids in motion,' Proc. Lond. Math. Soc, Wii. p. 103. 



