256 Prof. G. A. SSchott on Bohr's Hypothesis 



We easily find by Fourier's method 



n Z 2>eayGT sin ka. . / 



Comparing (22) with (8) we see that all the coefficients 

 vanish except C 3 (i> ^)? which takes the real value 



'deoovr sin ka/4:7r 2 a z k when^ — kco, 



but vanishes for all other values ofj. Using (13) and (16) 

 we find 



as— . 2 3 , -l 1 sin ka. . vrdixdz for j= —ka), 



= for all other values of j, 



whilst all the other coefficients b 3 , &c, vanish identically. 

 To evaluate the integral we put 



/»=(«* + />)* + *», g 2 =(*T-p) 2 + z 2 ; 

 neglecting higher powers of a/p we obtain from (21) and (22) 



_Jeo>p_ C> C^' An{W(€?-f)l P }gdfdg) 



Us -i^a%} J 2p _, ' V{g*-(f-2pY} 



em f sin7-7C0S7"l 

 = 4^i 7 J' where 7=Wf- 



(23) 



Since j=—ka), and all coefficients vanish except a 3 , we 

 find from (17) on putting cop = /3, the unit of speed being 

 still that of light, 



R = Trlaf P f '[/S 2 ! Jk'W sin <9)} 2 

 i Jo 



+ cot 2 . {J K (*£sin 0)\ 2 ] sin 6W. 



The coefficient of af in the sum is equal to twice the in- 

 tegral I x , introduced and evaluated by me elsewhere * with 

 l=zm = k; substituting its value and using (23) we obtain 



-P _ e 2 co 2 £ f Q si n 7 -7 cos 7 j 2 



[^ 2 J 2K '(2^)-P(l-/3 2 ) [^2K(2kx)dx\. (24) 



In order to convert to electrostatic units we must replace 



* * Electromagnetic Kadiation/ pp. 136, 137. 



