346-349] Collisions with Law of Inverse Fifth Power 291 



348. We require also to calculate AQ where Q is replaced by certain 

 functions of the second and third degree, but these quantities will be required 

 for a single gas only. In this case equation (686) becomes 



[Q]adade 



.(691). 



II. Q = u*. 



349. We begin by putting Q = u 2 . The value of w 2 u z for a single 

 collision, from equation (683), is given by 



u 2 - u z = u + (u' - u) cos 2 -^ - 5 VF 2 - (u' - u) 2 sin & cos eV - u 2 . 

 ( Z Z } 



If we substitute this value for [Q] in equation (691), and perform the 

 integration with respect to e, we obtain 



f / 6 9' \ 



I ( 2u(u'- u) cos 2 ^ + (u - u)- cos 4 ^ + 1 (F 2 - (w'-w) 2 ) sin 2 0'J ada 



(692). 



Since 



and 



F 2 - (u' - w) 2 = (v f - v? + (w - w) 2 , 



0' ff 



cos 4 = cos 2 -= | sin 2 6', 



_/ 



the bracket in the integrand becomes 



ff 



(u' z - w 2 ) cos 2 + {- 2 (w' - u? + (v 1 - v) 2 + (w' - w?} sin 2 &. 



Now 



M" - 



| - 2 (u' - u) 2 + (v' - v) 2 + (w' - wf | 



= | - 2 (u x - u) 2 + (v' - v) 2 + (w' - w) 2 1 

 = 2 (- 2ij2 + v 2 + w" 2 ). 

 Hence equation (692) becomes 



/ 



sin 2 0'ada ...... (693). 



Jo 



Maxwell writes 



TT sin 2 d'ada = A 2 

 o 



...(694), 

 192 



