118 Mr. G. H. Livens on the 



A slight and more suitable modification o£ this equation 

 is obtained by interpreting it in terms of <j> ; it is then very 

 easy to see how the various methods of approximation which 

 are usually adopted in this theory are effective in the final 

 result. 



If we therefore write 



f=<f> + ke-* u \ 

 the equation becomes 



(BV„)</>+(«V)4> + |f+^ 



= Ae-«" 2 j"2 ? (r ( R)-~ («V)A + m 8 («V)?], 



or using ^ = ' 2? ( M R)_ 1 ( MV )A+ M s («V)g, 



the equation for <£ is obtained in the form 





-qui 



This is a linear partial differential equation of a well-known 

 type. To obtain the general form of the function <f> which 

 satisfies it, we must first try to obtain seven integrals of 

 the equivalent differential relations 



Jf _ dt) _ d£ _ dx __ dy _dz _dt _ d<j> 



We notice, however, that six of these relations are the 

 same as 



-%—Y ^Q—Y ^—7 

 dt ' dt ~~ ' dt ~ > 



^_£ <%-„ &Z-r 

 dt ~ ^ dt ~ v > dt ~ & ' 



which are exactly the equations of motion of the typical 

 electron. The solutions of these six equations are obtained 

 at once, provided we know the values of (X, Y, Z) as 

 functions of the time and position. This provides us with 

 six integrals of the relations, and the seventh is got from 



# + r=X A <r""> 



dt Tn_ 



where, however, the function on the left is interpreted, by 



