289, 290] Persistence of Molecular Velocities 239 



and ON=$(OP+OM\ 



= \ (a + b cos #), 



= T- (3ft 2 + b 2 - r-\ 

 4ft v 



By differentiation of the relation (552), we have 



rdr = ab sin 6 d6. 



Hence the fraction (551) becomes 



ON** 



the limits of integration now being from r = a + btor = a~b. When a > b, 

 the fraction is equal to 



3o 2 + 6 2 _ 3^ /5ft 4 + 10a 2 6 2 + 6 4 

 ~~ba 20a ( 3ft 2 + 6 2 

 and when a < b, to 



3a 2 + 6 2 3 /56 4 + 10a 2 Z> 2 



,(553), 



+ b* 3 /56 4 

 la 20ft V 



36 2 + ft 2 



Expression (553) is equal to 



Expression (554) is equal to 



10ft (3ft 2 + 6 2 )' 



a (56 2 + 3ft 2 ) 



(555). 



(556). 



The essential point is that these expressions are necessarily positive for 

 all values of a and b, so that whatever the velocity of the second molecule 

 may be, the " expectation " of the velocity of the first molecule after collision 

 is definitely in the same direction as its velocity before collision. Expressions 

 (555) and (556) are too complex to convey much meaning to the mind. The 

 following table gives a, the expectation of velocity in the same direction as a, 

 corresponding to different values of the velocity b of the second molecule. 



-= i k f 1 

 a 



- = 500 -492 -473 '437 

 a 



- = 380 -368 -354 '343 

 a 



400 



00 



333 



