THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 203 
centred at O is cut by radii parallel respectively to the co-ordinate axes and to c 0 , r R , 
and c' R . Then we have 
(45a) e = C 0 C R C R , X 12 = C R Oc r . 
y 
It is convenient also to use certain spherical polar co-ordinates, as follows, taking 
Ox , 0 xy as initial line and plane for c R and c' R , and Oc R , 0 c R x, or Oc' R , 0 c' R x for c 0 . 
Thus we write 
(46) d R = c R Ox, 6 R = c R Ox, d 0 = c 0 Oc R , 6 0 = c 0 Oc K , A = c 0 Ox, 
(47) <p K = c R xy, <p' R = c' R xy, ^ = c 0 c R x, <p' 0 = c 0 c' R x. 
Evidently we have 
(48) cos 6 f 0 = cos 6 0 cos X] 2 + sin 9 0 sin x 13 cos e, 
(49) cos 6' r = cos 0 R cos xis+ sin 0 n sin xia cos e + 
(50) cos A = cos 8 0 cos @ R + sin 0 V sin 0 R cos <p 0 , 
— cos 6'u cos 6 ' R + sin 0' o sin 6' n cos <j>' 0 . 
Expressions for the Velocities After an Encounter. 
§ 4 (F) We have thus indicated how the final molecular velocities c\, c' 2 are to be 
determined (cf 43) in terms of the initial velocities ca, c 2 or c 0 , c R together with p and e 
(these being the eight independent variables of an encounter). This has been done 
by showing how c' R depends upon c R , p and e; it has in fact been shown that the 
spherical polar co-ordinates of c'r, referred to c R and the plane c 0 , c R as initial line and 
plane, are C R , X12 ( a function of p and C R ) and e. Hence we may at once write down 
