452 MATHEMATICAL APPENDICES 48* 



as the correct formula for the coefficient of viscosity, and I have 

 employed it in Chap. VII. in all the calculations of the numerical 

 values of the molecular free paths L. 



Even this calculation of the factor K depends, indeed, on 

 assumptions which are not exact, but only approximate. As has 

 already been mentioned in 47*, objection can especially be 

 raised that the law of distribution of speeds has been employed 

 in a form which is exact only for a constant flow, and that the 

 collision-frequency and the mean free path have been so taken 

 as if the gas were at rest. I cannot, therefore, gainsay Boltz- 

 m a n n ' s l expectation that another value of K would result from 

 stricter regard being paid to these considerations. But hitherto 

 no one has succeeded in overcoming the mathematical difficulties 

 of a more accurate calculation, and in bringing out from it a final 

 formula so simple that it can be turned to use for application to 

 observed magnitudes ; and, even if this were done, the result would 

 not practically differ from that given by the above formula, which 

 certainly does not depart much from the truth. 



49*. Friction Accompanying Change of Density 



In addition to the kind of internal friction that has been con- 

 sidered in the foregoing investigations, and which takes place in 

 like fashion in both liquids and elastic fluids, there can be ex- 

 hibited in the latter class of fluids a second kind of friction which 

 has hitherto not been experimentally investigated. The motion 

 does not undergo frictionlike resistance only when one layer slides 

 over or near another ; but such motions also as lead to compres- 

 sion or expansion are hindered by resistances, and these too we 

 may term internal friction, since they spring from the same 

 causes. These kinds of frictional forces which occur on compres- 

 sion or expansion have, therefore, been considered in all theories 

 hitherto put forward of the viscosity of gases ; and it has not 

 escaped notice that this second kind of frictional forces may 

 depend on a different coefficient of viscosity. For as the elasti- 

 cities of extension and tension require two different values of the 

 constant of elasticity, so also the friction that comes into play 

 with dilatation may be determined by quite a different coefficient 

 of friction from that concerned in the frictional phenomena 



1 'Theorie der Gasreibung,' Wiener Siteungsber. Abth. 2, 1880, Ixxxi. 

 p. 117 ; 1881, Ixxxiv. pp. 8, 1230. 



