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



The shadow produced by the sphere is therefore very 

 complete, and in fact much too complete to admit diffraction 

 as an explanation of the experimental results. For very 

 small orientations, of course, the exponential portions of the 

 sum could be of greater order than this, and they are not 

 included in this remark. 



The integration by parts remains valid for harmonics in 

 the vicinity of m = z, although R,„ and therefore ?«, is of 

 higher order in this case. For the change in R n is con- 

 tinuous, and it is only necessary to break up the integration 

 from e to cc into several stages in which different formulae 

 for R n are used. The continuity of the values of R K and 

 (f> n is demonstrated in a paper on the asymptotic expansions 

 of Bessel functions *. Since the term neglected in the inte- 

 gration is necessarily of the same order relatively to that 

 retained, in each part of the range, a series of integrations 

 by parts is sufficient to show that the terms near m = z are 

 already fully taken into account. Their main effect is in 

 fact exponential, as will appear later. This investigation is 

 sufficient to give an upper limit to the value of the diffracted 

 effect, and is complete enough to decide the main point of 

 controversy on this subject. It will be noticed that even 

 when points not on the surface are treated, and <f> n —$> nr is 

 therefore not zero, the effect cannot be of a greater order 

 than above, so that the use of a receiver not very close to the 

 surface cannot, with diffraction alone as an aid, furnish 

 an effect sufficiently great to be perceptible, for finite 

 orientations. 



But for other purposes, it is necessary to carry the calcu- 

 lation further, and to obtain an actual formula for the effect, 

 and it will be shown in the next section that the actual effect 

 is much smaller than the limit assigned above. 



Further e. rumination. 



We proceed to collect the terms of the order which is 

 apparently most significant in the expression for the magnetic 

 force. Since by (%$) 



■-=('+» 



neglecting # 3 and za?, therefore 



D ? / e = (^)(l + ^ 2 ) = cA, 

 Dh, e = 3/2* ; 



* Phil. Mag. Feb. 1910. 



