Energy Distribution for Natural Radiation. 795 



§ 3. Expression (ii.) may be regarded as the resultant of 

 two trains o£ waves of infinite extent, travelling in opposite 

 directions (parallel to Oy say) in the medium, with a velocity 

 C ; which velocity is that of radiation of frequency p, if the 

 medium is dispersive. If y = specifies the position of the 

 place where we are studying the value of X, we may con- 

 veniently represent X by 



X y=o= I %iO) cos [>(*+y/C) + a>i] dp 

 Jo y=° 



+ i X?(P) cos [^0-y/G) + fi> 2 ] dp. (iii.) 



Jo y=Q 



The first integral may be considered as representing a train 

 of waves stretching from y = to y = &> originally, and 



-> 

 travelling in the negative direction yO. The second in- 

 tegral represents a train of waves stretching originally from 

 y= — go to?y = 0, and travelling in the opposite direction. We 

 obviously get by comparing (ii.) and (iii.), 



XiG>) cos g>i + X 2 (?) cos © 2 = f(p) cos/3, . (iv.) 



X\(p) sin ft)i + % 2 (/') sin <o, = f (/>) sin (3. . . (v.) 



We may take (o x and <y 2 as being arbitrarily fixed, when 

 these two equations will give Xi(p) anc ^ X^P)- 



§ 4. We have now to consider the energy in the waves 

 represented by the expressions in (iii.). The energy per 

 unit volume due to X is eX 2 /8-7r *. If we consider unit area 

 at y = 0, normal to Oy, the amount of energy contained in 

 the waves given by (ii.) will be that corresponding to a 

 volume CT. We shall consider the energy E x ' corresponding 

 to the first integral (which we shall call X,) in (iii.). This 

 will be 



E/=4r x ' 2 ^> («•) 



where 



Xi=J xi(p)°° B l>(«+y/ c )+*i]i*- • ( viL ) 



To evaluate (vi.) we re-write (vii.) in the form 



J»oo 

 X i{q) cos b(*+y/C) +0)!] dq. . . (viii.) 

 o 



* e represents the dielectric constant of the medium for frequency p. 



3 F 2 



