Mechanism of Radiation. 435 



in the case in which r > i . Since %-/R) becomes identical 

 with E. -1 except for small values of r, we may assume 



when / ; >>-. where 6 X is a series in descending powers of r' 

 and ascending powers of r, which is very small except near to 

 the value i J = r. Similarly we may assume 



when i J <r. where e : is a similar series in ascending powers 

 of r* and descending powers of r. At r = /- , J C1 + 0*1) and 

 (l + e ly i must have a common limit (1-f £1). 

 In the identifies 



pnt/i(» i )=^(* / )( 1 +^)»/3( f 0=*(' l Xl+€i) and afld - We 

 obtain 



4- terms of lower decree in a. 



[ 



Jo 



- 



The function \n may be treated in the same way as r„. 

 except that the principal term r n /7 Jn+1 or r ,n /r n+1 is wanting. 

 We therefore assume 



r* 

 X n = -j^-{%2 between r I = r and x . 



X„ = - w — 1 e 2 between / J = and r, 



where 6* 2 , e 2 are similar to 8 l and e^ and have a common limit 

 f 2 at r~t>. 



Writing Y for x™ and Z for /SJ + invyf, we are now able to 

 expand equations (15) and (16) in the form 



+ terms of lower degree in rc, . . . (20) 



+ terms of lower degree in n. . . . (21l 



