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



This form must be used on the left of the transitional region. 

 Moreover, the derivates of the functions v are never zero. 



Magnetic force in the region of brightness. 



Proceeding first to the consideration of this special portion 

 of the problem, we note that the value of x defining the zero 

 point is given by 



x = r sin 0/R 



or in the figure, 



a? = sin OOP = sin*?!, .... (49) 



thus giving a simple law followed by the important harmonics 

 corresponding to any point P in the region. 



For the present, only a first approximation to the solution 

 is desired. More accurate values of the zonal harmonic and 

 of <p n would be needed if the calculation were to proceed 

 further. The second approximation will be given later. 



By a slight change of notation in (44), for this region, 



, 17T 



IZV+- 



? 

 n=l 



7^ = 2 ue 



n=l 



where 



u = -i«(l + «? r *»)(2c* sin <9R JC/aV)* ) 

 v = —cf) nr + $ n — m6. ) 



In the neighbourhood of the zero point, it has already 

 appeared that X n is negligible in comparison with unity, so 

 that e 2lX ™ = l, and we may write more briefly, 



u = -x(2zx sin 0R n U nr /a^7r) . ... (51) 



By (33), rejecting terms of order £ -1 in comparison with 

 those retained, if accents denote differentiations with respect 



