CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 



velocities among the molecules m.* Thus the aggregate value of 2)Q for the 

 < il.,- ill/ ,1: molecules m in a volume element dxdydz at the point considered is 

 given by 



I-'-! (",,, ', "). ( " ,,', '.', //') l>e the mean values of (u, >\ w), (', r', /). Except in 

 considering diffusion, we shall suppose that the mean velocities of each system of 

 molecules is the same, so that u = >/', ' = v 9 ', w = w '. We shall write U = u u^, 

 U' = tt'-ti,,', and so on, so that U = V = W = 0, U' = V"' = W 7 = 0. 



* 



The general equation of transfer of Q, independent of the action of external 

 forces ist 



here =r- denotes the "mobile operator" (-r + 'Wo +V -z- +w ;r-) of the hydro- 

 Dt \dt 3a5 9y tizl 



dynamical equations. The only term of the above equation which needs explanation 

 is AQ, which denotes the rate of change of Q due to the molecular encounters ; thus 

 the increase in 2Q which is produced in the element dxdydz by collisions \ in time dt 

 is dx dy dz dt AQ. The calculation of AQ is the immediate object of our investi- 

 gation. 



The motion of the mass centre G of two colliding molecules remains unaltered by 

 the encounter, and this point may be taken as the origin of a system of uniformly 

 moving axes, relative to which each molecule will describe an orbit in a plane 

 through G ; the two orbits will be similar to each other and symmetrical with respect 

 to the line of apses. If the molecules move with sufficient velocity to carry them 

 beyond the range of each other's action, the orbits will each have a pair of 

 asymptotes ; the asymptotes of the paths while entering on collision are parallel and 

 separated by a distance p (say), and the effect of the collision is to turn the direction 



* Thus the number of molecules m in a volume element dx dy dz which possess velocities whose three 

 components lie between u and u + du, v and v + dv,w and w + dw, is 



v /(, v, w) du dvdw dxdydz; 

 this property <Iefin?$ /(it, , w). Evidently 



III 



/ (u, , w) dudvdw = 1 ; 



f (, i', w) is, of course, also a function of /, //, ; and t in general. A similar function /' (u', r', w") exists for 

 the molecules m. 



t See JEANS' "Dynamical Theory of Gases" (Camb. Univ. Press, 1904), pp. 276-279; or BOLTZMANN'S 

 1 Vorlesungen iiber Gastheorie,' vol. i., 20. 



\ The terms collision and encounter are used indifferently to signify any mutual action of the molecules 

 which aflet-ts tlu-ir velocities. 



3 K 2 



