454 MATHEMATICAL APPENDICES 49* 



Our theory agrees, therefore, with all the older theories in 

 giving the value of the friction that accompanies alterations of 

 density as larger than that of the ordinary friction. The ratio we 

 have found of 3 : 1 is the same as that which, on Poisson's 

 theory, should hold between the elastic constants of extension and 

 torsion. In a theory of friction which I formerly developed from 

 other hypotheses 1 I found the same value for this ratio. Its 

 determination has, however, but slight practical value, since, as 

 indeed the last formula shows, this kind of friction gives rise to 

 forces which are not distinguished from the pressure, and may 

 therefore be reckoned in the value of the pressure. 



50*. External Friction 



The considerations and formulae of 47* at once supply the 

 means of determining the external friction which a gas experiences 

 at the surface of a solid body. 



Consider a gas which flows along the surface of a body at rest 

 and has everywhere the same velocity v ; then in each unit of 

 time a number of particles, which have the momentum 

 Q = yv = imN&v 



in the direction of flow, strike unit of surface and rebound from 

 it. Each particle rebounds from the solid wall with the same 

 speed with which it struck it, but not always in a direction 

 inclined to the wall at the same angle as that of the impact ; for 

 the solid wall, which is made up of molecules grouped together, 

 is, in respect of a striking molecule, an exceedingly rough surface. 

 Therefore the colliding molecules lose a part of their momentum 

 in the direction parallel to the wall, and this becomes transformed 

 into heat-motion. This loss appears as external friction, whose 

 intensity, therefore, is given by the expression 



where (3 is a numerical coefficient. 



1 Crelle's Journal fur Mathematik, 1873, Ixxviii. p. 130 ; with Addition 

 Ixxx. p. 315, with improvements by Stefan and Boltzmann. Other theories 

 of internal friction have been given by Navier (Mem. de VAcad. de Paris, 1823, 

 vi. p. 389), Poisson (Journ. de VEc. Poly. 1831, xiii. cah. 20, p. 139), Stokes 

 (Camb. Phil. Trans. 1849, viii. p. 287), Cauchy (Exerc. de Math. 1828, 3rd 

 year, p. 183), Barre de St. Venant (Comptes Rendus, 1843, xvii. p. 1240), 

 and Stefan (Wiener Sitzungsber. 1862, xlvi. Abth. 2, p. 8). 



