aC~ 



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



We recall from (83) that the magnetic force in the shadow 

 is given by 



y P = G{0)tm(R n R nr ) i (l + e 2, ^)sinm<£^-^ r ), (174) 



71-1 



where G(#) is an operation similar to g(6), and defined by 



2 i sin 2 6 d 

 ka 2 ir d0J 9 ^/2 ycostf — cosc£ 



and the exponents of the component series of yp have no zero 

 points. Since the integral formula for the zonal harmonic 

 is used, the result will not fail on the axis of the shadow. 



It was also shown, as in (89) and afterwards, that 

 when expressed as a function of m or n + ^, the function 

 mR n (l + e x n) is an odd function, and the conclusion was 

 derived that when R nr =R, n - and cj) nr = (j) m the harmonics of 

 low order in the series (174) cannot give a special contri- 

 bution of non-exponential type to yp, but must be included 

 in the sum subsequently derived by a use of Cauchy's 

 theorem of residues. 



The process of summation adopted would not, in general, 

 be useful in the presence of an exponent of argument 

 4> n — <f>nr which oscillates. We proceed to consider the 

 present problem by the same method, adjusting the analysis 

 where necessary. 



In the first place, as in (89), ~R nr is given by 



Rnr= \ Ko(2kr sinh t) cosh 2mt dt, . (176) 



7r Jo 



and this is an even function of m. The functions <p n and <j> nr 

 are not even in m, as may be seen, for example, by an 

 inspection of one of the forms of <£», namely, 



$> n = \l z 2 — m 2 *~ m sin " 1 m\z — mit\2 + 7r/4. 



But <f) n — <t>nr may be shown to be an even function. For 

 in general *, 



1 



[ (177) 



*•=-£ (s;- i )* ,+ * 



<j>nr=— I (v> l\Mr-\-kr — ^nnr — ^7r j 



where both R n and R nr are even functions of m. Thus 

 4>n — <t>nr is at once seen to be an even function of m also. 



* "The Asymptotic Expansions of Bessel Functions," Phil. Mag. 

 Feb. 1910, p. 285. 



