45* VISCOSITY OF GASES 443 



of the variables of integration, we get, on performing the integra- 

 tions, 



and thus by subtraction of these expressions the value of the 

 friction exerted 



dv , 1 d 3 v , 



The series obtained may be transformed into one proceeding 

 by rising powers of the molecular free path, or, more correctly, of 

 its square ; for by substitution of the value l/L for j3 it becomes 



and if the free path is really very small, this series will converge so 

 rapidly that we may neglect all the terms after the first, and write 

 for the friction 



F = %mNT ~ l L*(dv/dx)dy dz. 



The friction is therefore proportional to the surface dy dz on 

 which it is exerted, and also to the differential coefficient of the 

 forward velocity v with respect to x, the direction of the normal to 

 this surface. But this is Newton's hypothesis with respect to 

 the nature of viscosity, according to which it is taken to be pro- 

 portional to the relative velocity of the neighbouring layers, as has 

 been explained more at length in 74 ; for the relative velocity, or 

 the difference of velocity, of two neighbouring layers is expressed 

 by the value of the differential coefficient. Newton's hypothesis 

 therefore gives the viscosity as expressible by 



in which r? denotes the constant, which is called the coefficient of 

 viscosity. 



According to the theory just developed the value of this co- 

 efficient is 



for which we may write ( 75) 



7? = frnNGL or 77 = 



