297, 298] General Theory of Viscosity 245 



Since X is small compared with the scale of variation in Ji, this expression 

 may be written as 



JL ) - X,. cos 6 &} (564). 



This is the expectation of fi for any molecule which crosses the plane 

 z = z , having a relative molecular-velocity c inclined at an angle 6 to the 

 axis of z. 



The number of molecules per unit volume which have relative molecular- 

 velocities for which c and 6 lie within specified small ranges dc, dd may be 

 taken to be 



\vf(c) sin 6d6dc, 



f 

 where of course. I /(c) dc = 1. 



J o 



The number of molecules having a velocity satisfying these conditions, 

 which cross a unit area of the plane z = z in time dt, is equal to the number 

 which at any instant occupy a cylinder of base unity in the plane Z = Z Q and 

 of height c cos ddt, and is therefore 



^ vcf(c) cos 6 sin Qdddcdt. 



Each molecule, on the average, carries with it the amount of momentum 

 given by expression (564). The total momentum transferred across unit 

 area of the plane by the molecules now under discussion is therefore 



%vcf(c) \Ji (z ) X cos tffx'-) \ cos 6 sin ddddcdt, 



( \v z ' z=s<t) 



where X is the mean value of \ r for all such molecules. If we integrate this 

 expression with respect to 6, the limits being to ?r/2, so as to include all 

 directions in which the plane can be crossed from below, we obtain 



- IX (If) \dcdt, 

 \vz/g =z<> ) 



and if we now integrate this with respect to c, remembering that 

 f(c)dc = ~L, we obtain 



I 



J 



}dt .................. (565), 



<,) 



where c is the mean value of c, calculated as explained in 131, and Xc is 

 the mean value of Xc. We have seen that in general X is a function of c, and 

 have calculated the dependence of X upon c when the molecules are supposed 

 to be elastic spheres. At present we are not making any assumption as to the 

 structure of the molecules, and so cannot carry the calculation of the term Xc 

 any further. 



