252 Prof. G. A. Schott on Bohr's Hypothesis 



12. It is not difficult to reduce the Bessel Function 

 integrals in (15) to simpler forms, but the results are 

 complicated and not easy to interpret. Fortunately this 

 reduction is not necessary for our purpose owing to the 

 smallness of the electron. If p denote the radius of the 

 circle described by the centre, and a a length comparable 

 with the linear dimensions of the electron, for instance its 

 radius if assumed spherical, which is, however, not neces- 

 sary, then the coordinates -or and m' differ from p by 

 quantities of the order a, and z — z' is of the same order of 

 smallness. Thus, if we suppose the integrand of (15) 

 expanded in a series of terms of increasing order of small- 

 ness, the first term will determine the sign of the radiation R, 

 unless it should happen to vanish. To find this principal 

 term we need only put w and tv' each equal to p and z—z' 

 equal to zero in the Bessel Function and cosine terms 

 respectively, but we shall refrain from doing this in the 

 coefficients A 1? &c, because we know nothing as to their 

 form. For the sake of brevity we shall write 



a 1 (j,k)=^A 1 (j,k)d^dz=^A 1 '(j,k)d^'dz' . (16) 



for all values of j and k, with similar expressions for the 

 integrals of the remaining coefficients. Then we find to a 

 first approximation 



R = i7rSS 



I [_ j («3 + h cos 0)jpJ K '(jp sin 0) 



(. n m % cos — b 2l \ T/ . . „ . 1 

 7n, j p sin + sin e kj JkO/> sm 0) J- 



+ j (& 3 + »h cos 0)jp j k Up sin 0) 



lo cos — a 9 



+ { ('3 ~ «2 cos 0)jp3 K '(jP sin ff) 



+ ( i]ipS in^ + ^p^)jKO>sin^)} 2 



+ j (%- h cos 0)jpJ K '(jp sin 0) 



( . . „ a 3 jp cos + l 2 7 . T , . . /in 1 2 1 

 -(^p sin + 3 ^ sin e kjJ K (jp sin 0) j ^ 



. sin 0d0 (17) 



