78 BEPORT— 1881. 



generally received, A =:; 3 B. O. E. Meyer ' (1861) assumes tliafc the 

 friction on a small plane in a given direction in the plane is proportional 

 to the rate of variation perpendicular to the plane of the component of 

 velocity in the given direction, whilst there is a normal part, proportional 

 to the rate of variation perpendicular to the plane, of the component per- 

 pendicular to the plane. This is not so stated in his preliminary hy- 

 pothesis, but is vyhat his initial expressions imply. Considering then a 

 small parallelepiped dx dy dz, he arrives at the above form of equation if 

 B is put zero, which agrees with the results found by previous investiga- 

 tors for incompressible fluids only. Stefan ^ (1862) applies directly the 

 methods of elasticity to a small tetrahedron treating the forces as functions 

 of the nine differential coefficients of u v w, and shows, as in the theory of 

 elastic solids, that the forces are of the form — 



He now attempts to find a relation between X and fx on the following 

 hypothesis. Drawing a small plane through the direction of the velocity 

 at any point, then the friction must fall in this plane and be parallel to the 

 direction of the velocity. From this it results that A. = o. Stokes' cor- 

 responding assumption was that a uniform expansion of any element does 

 not require a re-arrangement of the molecules, which leads to X = — 2 ^/3. 

 Stefan's equations then give, in the typical form, B = o. From a totally 

 different point of view has Maxwell ^ approached the subject when the 

 fluid in question is gaseous. He bases his theory of viscosity on the 

 transference of momentum from one layer to another by the moving 

 particles of a gas, treating a gas according to the kinetic theory. He 

 obtains precisely the same expressions as Stokes. But this investigation 

 of Maxwell's is far more important from another point of view, in that he 

 attempts to express the unknown constant of internal friction in terms of 

 known properties of the meditim. The first attempt towards this was 

 made in his first paper'* on the kinetic theory of gases (1860), where on 

 the same bases as to the cause of friction he calculates the constant on the 

 supposition that the atoms of a gas are spherical and perfectly elastic, 

 with the result that the constant is proportional to the square root of the 

 absolute temperature and is independent of the density. O. E. Meyer ' 

 (1865) also arrived at similar results from the same data. But this does 

 not agree with experience, and Maxwell returns to the question again in 

 the paper already mentioned. The fact that the coefficient of viscosity 

 is independent of the density follows, whatever be the law of repulsion 

 between the particles, but the law of its variation with the temperature 

 depends on the law of force. Maxwell has chosen the inverse fifth, from 

 which it results that the coefficient is directly proportional to the absolute 

 temperature. Maxwell's result in this case is that A = Jcp/p where Jc is a 

 constant depending only on the mass of a molecule and the force between 

 two molecules at unit distance. He has also given an expression for the 



1 ' Ueber die Eeibung der Fliissigkeiten,' Jiorc?t, lix. p. 229. He acknowledges 

 in 1874 {Bnrch. Ixxviii. p. 131) that the investigation is not general. 



2 'Ueber die Bewegung fliissiger Korper,' Siti. Alad. Wiss. Wicn, xlvi. p. 8. 



3 ' On the dynamical theory of gases,' Trans. Roy. Soc, 1867, p. 81. 



* 'Illustrations of the dynamical theory of gases,' Phil. Mag., 1860, January and 



July. , „ 



^ ' Ueber die innere Reibung der Gase,' Fogg. Ann. cxxv. 



