260 



Free Path Phenomena 



[CH. XIII 



...(587), 



From equation (402) we have the relation 



_1^5 



v ~ JmdT ' 



where C v is the specific heat at constant volume, and again, from equation 

 (572), if K is the coefficient of viscosity, 



K = jfitclm (588). 



Hence, by comparison of equations (586), (587) and (588), 



v rd ^89^ 



J K\J V \<)<3O ). 



Correction when Molecules are Elastic Spheres. 



313. We found that the first formula obtained for the coefficient of 

 viscosity was true as regards order of magnitude, but required correction by 

 multiplication by a numerical factor substantially different from unity. So 

 also here, we shall find that strict analysis leads to a value of ^ which differs 

 very appreciably from that given by equation (589), although the only 

 difference lies in the occurrence of a numerical multiplier. We proceed to 

 apply analysis as rigorously as possible, to the case of conduction of heat in 

 a gas of which the molecules are elastic spheres. The solution which follows 

 is substantially that given in Meyer's Kinetic Theory of Gases. The main 

 difference is that Meyer neglects certain terms expressing the variation of 

 collision-frequency, although these terms are of the same order of magnitude 

 as terms retained. I have found it possible to give the more complete 

 investigation in which these terms are taken into account, but find that 

 their ultimate effect on the result obtained by Meyer is nil. 



z=z n 



314. We consider any element dxdy of the plane z = z 0> and with the 

 centre of this element as origin, we take 

 spherical polar coordinates r, 0, </>, the line 

 6 = being parallel to the axis of z. 



The curvilinear element of volume for 

 which r, 6, <j> lie between r and r + dr, 

 6 and 6 + dO, </> and < + d(f> is the volume 

 dv= 



We begin by considering the possibility 

 of a molecule undergoing collision in the 

 element dv, leaving it with a velocity c in 

 such a direction as to pass through the 

 small area dxdy, and describing a free path 

 which reaches at least as far as the element dxdy without collision. 



Since the whole motion is reversible^ the number of collisions in which 

 one of the molecules has a velocity between c and c + dc after collision in the 



