the Radiation from Electron Orbits. 649 



case of alb very small. The equation becomes 

 WK 



and its solution is K = Ae -6 where A is a function of a only. 

 Similarly K' = AV 6 , where A' is the same function of —a. 

 Hence we have 



I 

 J: 



cos ax cos (ft tan %) dx = i (A + A') £ -6 , 

 sin ax sin (6 tan x)&%— ^(A'— A)£~ 6 . 



It follows that I and J are each of the form 



G*/(/3)*-^ a " lcosec £, .... (28) 



and that I 2 + J 2 is of the form 



I2 + J2 = GF(/3)^- 2 w lG " lcosec / 3 . . . . (29) 



9. The value of H 8 is now jucot 2 /3, so that 



HJH = /u cot /3 cosec 2 /3, 



aud expression (18), which gives the number of orbits per 

 unit time of given class, becomes 



4tt 2 NA*- ag > cot j3 cosec 2 dG d/3. 

 Hence 



F p = 8 ^|^ f *' f G ,-*a-^ia-i cosec.3 ^ rf/8 dG _ 



Jo Jo 



Integration with respect to G requires the evaluation of 

 an integral of the type 



st on 



e -aG-bG-\ lQ ^ .... (30) 



-r 



the required integral being —dyjda. 



It is easily found that y satisfies the differential equation 



tt l y _a 



"db 2 ~- b y ' 



of which the solution is y = AafiK^iafi), in which a? = 4a£>, 

 and A is a function of a only. Also from equation (30) it is 

 clear that ay must be a function of ab only, and therefore a 

 function of x. 



