Mechanism of Radiation. 429 



The potential arising from the displacement of the whole 

 atom will therefore be 



E= JJK (r_f) ^ +(v '~ v) W + (? '~ ?) 5? }xi(W^'<¥d-, . 



The integration extends over the whole atom, but it will 

 be convenient to take the integration throughout all space, 

 writing p = at points outside the atom. 



This potential may be at once transformed, so that we 

 obtain 



E = _ 'I!J{&' m + §f (v ' pl) + §? (?M } xi(R) dx ' dyl dz ' 



Returning to polar coordinates, we can write E = Ex + E 2 , 

 where 



/ 2 sin ff dr' tiff d4>'. 



E -- - ( u h + "&> + ,c -k )j]j* i(R)pV2 5in ff dr ' cW *#> 



For any displacement which is continuous at the poles, 

 i'/sin 6 must be finite at both poles, and we may therefore 

 assume expansions in the form 



u=SS<PjO»). 



,im$ 



sip {v sin ^)=ssfl:prw^ 



w=SS7fPjW^ . . . . (1) 



where /z, as usual, stands for cos 6 ; n has all values from 

 to oo, and m from —n to -t- n, P~'"(/i) being taken equal to 

 P™(aO* The symbols «;;', /3"\ 7™, denote functions of r, and 

 the letter m is not an index but is written from analogy with 

 P^(yLt). The expressions on the r.h. sides of (1) are capable 

 of representing the most general displacements which are 

 physically possible. 



Substituting these values. Ei can be written in the form 



F™(li')e Sm v'r' 2 sin & dr' d6' dj>'. 



