hetween Radiation and Free Electrons. 19 



Let Cq stand for u 2 -f v 2 -f to 2 , and let the mean-square 

 velocity of the electron, averaged over a great length of 

 time, be c 2 . Let the proportion o£ the whole time during 

 which the velocity components lie within a small range 

 ducdvndwn. be 



a/( v+ ;' + ^ ^ 



Then the total radiant energy emitted by the electron per 

 unit time will, on the average, be 



^f( Uo2 + V ^ W ° 2 ) I {0, f) sin 6d$ dyfr du dv dw , (5) 



the integration being over all values of 6, ty, u Q , v , and ic , 

 and the frequency of any element of the light being given 

 by equation (4) . 



To analyse this radiation according to frequency, we may 

 change the variables from u , v , iv , 0, and ty to w , io , 0, 

 AJr, and q. Writing q for /eV, the frequency of the incident 

 light, we have 



If q V 2 ~] 



^ = sinflcps^ V ~"° C ° S °~ W ° Si ^ Sin *~ & V+^oJ 



whence integral (5) may be written in the form 

 Al dq \\\\f\ 72 (V + w o 2 ) + ~2 cosec 2 sec 2 -^ 



x ( Y— «. cos 6 — ^(' sin (9 sin i|r— — -4= ■ } > 



\ ?o(V + w )/J 



1 V 2 

 X I (0, ^) - /y_i_ y cosec sec ^ dO dty du div , . (6) 



•of which the form after integration is 



JXl- c2 >* w 



4. Suppose the electron is in a region of space in which 

 the law of partition of radiant energy is <j>(q )dq , the energy 

 being distributed at random as regards direction. Suppose 

 that unit energy of frequency q is, as the result of interaction 

 with the electron, after unit time replaced by energy #(y ) 

 of the original frequency q , and a spectrum \ F(q , q) dq of 



C 2 



