536 Dr. J. W. Nicholson on the Bending of 



If u = © 2 , 



I = (^iy^ da exp . t (_J*_«R + ^ sin 6<o>), (62) 



and I is expressed as a linear combination of FresnePs 

 integrals. This result holds for all points within the region 

 of brightness, for 6 has not been assumed small. 



dl/dd vanishes with 0, so that the axis of the oscillator is 

 a line of no magnetic force. This could have been foreseen 

 by considerations of symmetry, for the oscillator alone can 

 produce no magnetic force at points in its axis. 



Except when 6 is practically zero (in which case the 

 magnetic force becomes evanescent), the upper limit of the 

 integral may be treated as infinite, on account of the rapid 

 oscillation of the integrand. 



Now by well known results, 



I cos Xo) 2 dco = (7r/8\)a = + j sinXo> 2 da>, 

 Jo Jo 



that 



^exp.^-iTr-Ht-^sin^-^^^)* 



«-* B 



SO 



j; 



Thus _ / «T \i ■ M 



l ~ \2kaW) 6 ' * • ' ' (b ^ 

 and since ^ ,~ a . am 



the leading term of BI/d# is 



(7rto72R 4 )i sin 0*-'* R , 

 leading to 



A 



7/d= -2ik^ sin 2 6e- lkR , .... (64) 



as for the points more remote from the axis. There is thu 

 no change of type in the solution near the axis. 



Extent of the region of transition. 



An estimate of the extent of the region hitherto called 

 transitional may be obtained from the consideration that it 

 is the region in which the expressions, previously used for 

 the Bessel functions in the calculation of the significant 

 harmonics, cease to be valid. 



