178 Radiation from an Electric Source. 



maxima of radiation will, by (25a), be given very nearly by 



max. ~" aw* ~~ 4:TT 2 a\aJ ^~ 



They will be denoted by J l5 J 2 , &c, and the corresponding 



wave-lengths \ by \ l5 \ 2 , &c. They corrrespond to cos 2 17 = !, 



i. e. tan 77= ±00 , and are therefore roots o£ the transcendental 



equation 



cos u K A /OON 



it? tan ^ = —7-. +^=0, . . . . (29) 

 g(u) ^ u 2 



as mentioned in the first Note. With the only exception of 

 the nearest neighbourhood of these roots, we can write 

 cos 2 v/w 2 = (iu 2 + id 2 tan 2 77) _1 = (w tan 2?)~ 2 , that is, instead 

 of (25a) } 



a \^(2() wv ■ a 



because K/2t 2 is (everywhere but at i(i, u 2i &c) Zar<7<? * in 

 comparison w T ith cosu/g(u). For the same reason, as is seen 

 at a glance from (29), the positions of the maxima are very 

 near the corresponding zeros of radiation, i. e. 



\i, A 2 , &c. are slightly > vi, v 2 , &c, 



the differences being almost imperceptible. Numerical 

 examples will be given in a later note. To put it briefly, the 

 spectrum, for large K, is practically confined to very narrow 

 bands with J max . proportional to (\/a) 2 , the intensity ot 

 radiation elsewhere being proportional to (a/X) 2 . As a rule 

 (to which exceptions may occur in connexion with possible 

 peculiarities of K as a function of \), the passage from light 

 to darkness is more abrupt towards the ultra-violet than 

 towards the infra-red, although it is, for all purposes, abrupt 

 enough on both sides. The position of these sharp spectrum 

 " lines w is practically given hy the roots Mj, u 2 , &c. themselves. 

 The promised justification of contemplating large permit- 

 tivities at all, and enormous ones especially, will be given in 

 the next Note. 

 May 5, 1915. 



* Even for high values of u (viz. m), for these occur only iu the 

 neighhourhood of a convergence-point or " head," where K is very 

 rapidly increasing. 



