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



function of m, and may be expanded in an asymptotic series 

 in the form 



mA + m 3 A x + m :, A, + ... 

 so that 



yp = G(0) %* (wA +w* 8 A 1 +.:.) sin mfc 



where the coefficients A are independent of m. 



Now it is known that, when convergent, the series 



I m 2 ? +1 sin«i0 = (90) 



»»=0 



if p is an integer. 



Now near the lower limit e of the integration in the 

 previous section, it is known that the coefficients A decrease 

 in order of .:, from the form of R„ as a function of m when m 

 is small. It follows that the terms of yp expressed as a series 

 in descending powers of z will continue to vanish, and this 

 shows that the actual value of yp must be of an exponential 

 form for points on the surface of the sphere. Thus the result 

 of M. Poincare's revised investigation, and that given by the- 

 method of this paper, are not at variance for surface points,, 

 and the effect in this special case is entirely exponential. 



But this line of argument is liable to failure when the 

 magnetic series contains an oscillating exponential of argu- 

 ment proportional to <£„— </w, and it is therefore not yet 

 shown that the effect at other points in the shadow is deter- 

 mined by an exponential law. M. Poincare's mode of proof also 

 fails, in its present form, for other than surface points, as it 

 definitely assumes the absence of oscillation. 



In the next section, a preliminary discussion of the 

 exponential sum is given, and the investigation for other than 

 surface points is postponed for the present. 



Determination of the e.rponential sum. 



As the sum of the harmonic series, for points on the surface, 

 is now shown to be mainly caused by terms in the neighbour- 

 hood of singularities of the function u, it is simpler to proceed 

 otherwise at this point, as a direct summation may be effected. 

 Let v be a typical value of m making 



d/tk. -«,„(-) = (91) 



Then m = v is a pole of the function u, and this must be a 

 simple pole because u is proportional to the ratio of z*K m (z) 

 to its derivate. Again, on reference to the original expansions 



