250 Prof. G. A. Schotfc on Bohr's Hypothesis 



11. We now substitute from (12) in (7) and integrate with 

 respect to t-i from to T and with respect to <£ from to 27T.. 

 All terms vanish except products of terms of U^ and U<p in- 

 volving reciprocal time factors exp it and exp(— it), and 

 these acquire the factor 27rT, which cancels out from (7). 

 From (9) we see that these pairs of terms correspond to 

 equal and opposite pairs of values of j and k ; clear!}' the 

 reality of the motion requires that the coefficients C l5 &c.,, 

 belonging to such pairs should be conjugate imaginaries. 

 Moreover, the signs and magnitudes of the functions 

 Jk{J^ sin 6) and J V (,;«■ sin 6) both remain unaltered when 

 the signs of bothy and k are changed. Under these circum- 

 stances the complex integrals in (12) change into their 

 conjugate imaginaries, and each integral when multiplied by 

 its conjugate gives a term of R. 



In order to express these terms explicitly we write 



1 , ,*)=A 1+ »B, CxC-i, -*)=A 1 -»B b \ 



K,0', *) = 14 + ^1,, K.i-j, -*) = L 1 -*M 1( / • W 



with similar equations for the remaining coefficients. Then 

 the double integral in the first equation (12) becomes 



J)[{ 



(A 3 -f- L 2 cos 0)j^J- K ! (jsy sin 6) 

 M a cos0-B< 



/ . . M 8 cos V — b 2 \ • n\\ 



-(M^grsinfl-f- gin ^ kj J K 0^sm(9)j 



+ 1 1 (B 3 + M 2 cos 0)JvtJk'(Jvt sin 6) 



+ (L, j*? sin 6 .+ 3C ^ d ^" 2 *) Jk(;<* sin (9) | ] 



. exp ijz cos 6 . dvrdz (14) 



For the conjugate we must of course change the sign of c, 

 but in addition it is convenient to replace the variables of 

 integration, z, -ar, by z\ w', the coordinates of a second 

 element of the electron, and to write in place of the co- 

 efficients Ai, &c, tbe corresponding functions Ax', &c, of 

 the new variables z', ot ; '. 



On multiplying corresponding conjugate integrals together 

 we obtain a fourfold integral with respect to the four vari- 

 ables z, ot, z', and «r', the integrations with respect to z and 

 <G7, as well as with respect to z' and -or', being extended over 

 the area of the meridian plane swept over by the electron in 

 its motion. This fourfold integral is itself complex, but the 

 conjugate integral is obtained by changing the signs of 

 the parameters j and k, and the two fourfold integrals 

 together contribute their real part only to the radiation. 



