45* VISCOSITY OF GASES 441 



as before the radius of the sphere of action and the mean dis- 

 tance between neighbouring particles. Putting for r, which is 

 still undetermined, the distance of the point (x, y, z) from the point 

 (a/, y>, O, or 



r 2 = (x r - x? + (y f - y)* + (z' - z)\ 



we may interpret the magnitude 



NT ~^-^ dx'dy'dz' 



as the number of particles which start from the element dx'dy'dz' 

 in unit time and traverse a sphere of radius r described about the 

 element as centre, so as to cut the surface-element dy dz. From 

 the number of particles traversing the whole spherical surface we 

 deduce the number of those crossing the element dy dz by com- 

 paring the projection of the element on the spherical surface with 

 the area of the whole sphere. The latter amounts to 4?rr 2 , and the 

 former to dy dz cos s, where s denotes the acute angle which the 

 direction of r makes with the a?-axis. The number of particles, 

 therefore, which in unit time reach and pass through the element 

 dy dz, having started from the volume-element dx'dy'dz', is 



NT-V^Trr 2 )- 1 cos s dydzdx f dy f dz f . 



The next question is, How much momentum is carried over 

 by these particles ? Since the molecular motion, of which heat 

 consists, is taken to be the same throughout the medium, its 

 transference causes no change, and it may therefore be left out of 

 account, and we have to consider only the forward motion of the 

 layers. Let this occur with velocity v at the point (x, y, z), and let 

 v f be the corresponding value of this function at the point (x f , y', z'). 

 Then the momentum leaving the element dx'dy'dz' and crossing 

 the element dy dz in unit time is 



dQ = (m/^NT-We-^r-* cos s dy dz dx'dy'dz'. 



From this, by integration with respect to x', y', z' over one-half 

 of the medium, we obtain the total value of momentum carried 

 over from this half of the medium through the element dy dz of 

 the dividing plane into the other half. If we take the medium 

 as unlimited, this quantity which is carried over in the direction 

 of increasing x is 



Q l = dy de(m/4nr)NT ~ l f dx' f dy' f dz f e- ffr v f r~ z cos s, 



Jx J-co J -co 



