IONS PEODUCED BY RONTGEN RAYS IN GASES AND VAPOURS. 273 



force, taken as wholly radial, between the ion and the molecule is given by dR/dr at 

 a distance r. Let us reduce the molecule to rest and consider the relative motion as 

 the ion approaches it so that the velocity at infinity was U. The interacting forces 



must now be considered as being derived from a potential ]F^f3&fn denoting the 



/**** /yq 

 mass of the molecule. X*s4&*^ 



The shortest distance r to which the ion and the molecule approach is given by 



the equations 



= ru, 



m 



where u denotes the velocity of the ion in this position and b is the length of the 

 perpendicular from the molecule to the rectilineal path of the ion. 

 We deduce 



A collision will take place if 



where cr = ^s' + ^s, the sum of the radii of the force spheres of the ion and molecule, 

 and 11, denotes the value of E, at collision. 



If R = 0, i.e., if the polarisation of the molecule due to the ionic charge is negligible, 

 the condition for a collision reduces to b S cr, as is otherwise obvious. 



The connection between the velocity U and the mean thermal ionic velocity V can 

 be deduced by the application of MAXWELL'S law of distribution of velocities. We 

 obtain 



U 2 = VU+ N 

 m 



Hence A ^- U 2 = AMV 2 = mv 2 ; thus A -- U 2 is the mean kinetic energy of 

 2 M + m 2 M + m 



the molecular motion. 



The effect of the polarisation due to the ionic charge is, therefore, as far as collisions 

 are concerned, to replace <r 2 by cr 2 {1 + 2R <7 /mv 2 }. 



Now the mean free path of an uncharged body of the same mass and dimensions as 

 the ion is given by {ira<r 8 \/l + M/m}~ 1 , where n denotes the number of molecules per 

 cubic centimetre. Hence the actual mean free path of the ion is L, where 



VOL. ccix. A. 2 N 



