ARISING FEOM INEQUALITIES OF TEMPERATURE. 687 



Let the masses of the molecule be M l and M 3 , and their velocity- 

 components , fy, u and ,, -r),, 4 respectively. Let V be the velocity of 

 M t relative to M r 



Before the encounter let a straight line be drawn through M 1 parallel to 

 V, and let a perpendicular 6 be drawn from J/ a to this line. The magnitude 

 and direction of b and V will be constant as long as the motion is undisturbed. 



During the encounter the two molecules act on each other. If the force 

 acts in the line joining their centres of mass, the product bV will remain 

 constant, and if the force is a function of the distance, V and therefore b 

 will be of the same magnitude after the encounter as before it, but their 

 directions will be turned in the plane of V and b through an angle 20, this 

 angle being a function of b and V, which vanishes for values of b greater 

 than the limit of molecular action. Let the plane through V and b make an 

 angle <f> with the plane through V parallel to x, then all values of (j> are 

 equally probable. 



If ,' be the value of , after the encounter, 



M 

 When the two molecules are of the same kind, ,, V> = \, and in the 



present investigation of a single gas we shall assume this to be the case. 



If we use the symbol 8 to indicate the increment of any quantity due 

 to an encounter, and if we remember that all values- of <j> are equally probable, 

 so that the average value of cos <f> and of cos 3 $ is zero, and that of cos 5 (f> 

 is , we find 



(+) =o ................. . ......................... .................. (2) 



(3) 

 ............. (4). 



From these by transformation of coordinates we find 



l )sm'e<x>B>0- ......................... (5) 



- 3 (tf + tf) 

 7^ +F*)] sin' cos' ............... (6) 



3 (M + &7,t) = - i [9 (M + &?,Q - 3(frfc& + M + f fl + &7,& 



.)] an 1 6 cos 2 ...... ...(7). 



