288 Free Path Phenomena [CH. xv 



If we suppose the angle XGY to be &> 2 and XGZ to be <w 3 , we shall have 

 in a similar way, 



v' - v = (v' - v) cos 6' + \f F 2 - (v - v) 2 sin 6' cos (e - &> 2 ) (679), 



w' - w = (w' - w) cos 6' + V F 2 - (/ - w) 2 sin 0' cos (e - &> 3 ) . . .(680). 

 To determine eo 2 , &> 3 , we notice that in the triangle GXY, XY = -, 



m 



XGY = ft> 2 , hence 



(u f - u) (v' -v) + V( F 2 - (u' - u) 2 ) ( V 2 - (v - v) 2 ) cos &> 2 = . . .(681), 

 and there is a similar equation for co s , in which w' w replaces v' v. 



We are engaged, it must be remembered, in calculating the changes 

 in the velocities which are caused by the collision only, so that the changes 

 caused by the action of external forces, if any, may be neglected. We may 

 accordingly use the equations of conservation of momentum. Thus 



m^u + m 2 u' = mjU + m z u' (682), 



an equation which can of course be obtained directly, by the integration 

 of the equations of 340. 



Multiplying equation (678) by ra 2 and subtracting from this the equation 

 of momentum, we obtain 



(m-L + w 2 ) u m^u + raX m 2 (u' u) cos 6 7n 2 V F 2 (u u) 2 sin 0' cos e, 

 so that finally 



u - u = ^ |2 (u' - u) cos 2 - - \/F 2 -(w'-w) 2 sin & cos el . . .(683). 



This gives the change in u at a collision of the type we have been 

 considering. The changes in v and w are given by similar equations, and 

 from these we can calculate the change in Q, when Q is any function 

 of u, v, and w. 



345. Before discussing the frequency of collisions of given type, we must 

 make a remark as to the definition of the commencement of a collision. The 

 law of force assumed is such that two molecules are influenced by one 

 another at all distances, but the influence only becomes perceptible when 

 the molecules are near to one another. If we suppose the influence to 

 be perceptible when the molecules come within a distance a- of one another, 

 we may regard <r as the diameter of a sphere of molecular action, and may 

 define the commencement of a collision as the instant at which either 

 molecule crosses a sphere of radius a- surrounding the other. This gives an 

 approximation which is the better, the greater the value of a. In the limit 

 we shall find it possible to pass to the limit and take a = oo , in this way 

 disposing entirely of the error involved in assuming the existence of a sphere 

 of molecular action. 



