290 Mr. G. H. Livens on the Electron 



then 









a-&= J 



r — To 

 / > 

 T m 







and the equation 



for / is thus 











y|/ + 



ov 



3? d# 





f-fo 



y 

 Tm 



of which 



the general 



solution 



appropriate 



to the 



present 



type of problem is 

 where 



A B| ov o? o^ oy b d* d* 



and the suffix x indicates that the functions expressed ex- 

 plicitly as functions of the general time variable t are to be 

 taken for the time t — t^. 



We now require some idea of the value of r m in the par- 

 ticular case under investigation : this is obtained at once 

 from an investigation of the dynamics of a collision between 

 an electron and an atom. 



We introduce polar coordinates in the plane of the motion 

 of the electron and the centre of the atom, with the origin at 

 this centre and the axis along the direction of the asymptote 

 in the direction in which the electron approaches the atom. 

 The two first integrals of the equations of motion of the 

 electron are then of the usual type, 



r 2 6 = h, 



-$ 



where C and h are constants. If u is the velocity of 

 approach from infinity and p the perpendicular from the 

 centre of the atom on to the asymptote described with the 

 velocity w, then 



h=pu i C = ?i 2 . 



If we eliminate the time differential from the two equations 

 of motion and substitute the value of the constants, we get the 

 equation to the path of the electron in the form 



f/dr\ 2 f = A 

 r 4 \de) r r*u 



+ 1, 



