286 Dr. J. TV 7 . Nicholson on the Bending of 



the possible error is much less, and by a method used in the 

 case of the shadow round the sphere, it may be shown that 

 the contribution to the magnetic force made by this neglected 

 series, in company with the other series which has no zero 

 point, continues to vanish term by term. 

 We therefore write finally 



S=4* e' zv " f° rfr»(v e +V 2 |j + ... ) i z -\ . (123) 



and only the first two orders in z will be retained. To these 

 the above proof applies. 



The integrals concerned here are well known. Quoting 

 their values, 



S = 2(«)»|V - £}«•"!.+*•* . . (12^ 



to a second approximation. If vj were negative, the sign 

 of \iir would be negative, and in V , V 2 as calculated later, 

 (VO* would become ( — «</')*. 



Calculation of the functions V , "V" 2 in general. 



The calculation of V and V 2 is somewhat laborious, suc- 

 cessive stages being shown below. We recall that if U is 

 defined as a function of: x as in (116), then V=U cojv f ex- 

 pressed as a function of (o. V is the value of this function 

 at x = x , an d V 2 is its second derivate with respect to ay at 

 this point. Moreover, the relation between &> and x is 

 w — x/v — Vq, v being a function of x, so that x = x corre- 

 sponds to &) = 0. The suffix zero applied to any quantity 

 denotes that its value at the zero point must be taken. 



Let y = x—x > so that B/B#=d/d# ? and let either of these 

 operations be denoted by the accent. Then, v ' being zero, 



U^U +»Uo'-+gu."+„. 



2! 



and on redviction, v " being positive, and a> having the sign 

 of y by its definition, 



a>=(t-t< )* = y^r / y2{l + Ay + By 2 + ...}, (125) 



