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



given effectively by its first term. Successive terms would 

 not decrease exponentially, and a process of summation 

 would be required for this series. This would involve 

 difficulties. 



But we have shown the absence of such zero points within 

 the shadow, and therefore the absence of this special 

 difficulty. It is now evident that in the present problem, 

 a determination of a v for the first singularity will complete 

 the physical solution. But it is also evident that a similar 

 treatment would fail for points not inside the geometrical 

 shadow. 



The determination of a v may be made at once. For it 

 will differ from its value for points on the surface, only by 

 a factor 



(R nr /R„)V^-M, .... (181) 



provided that we substitute, in this factor, the value of m 

 given by 



where (3 has its previous value '696. 



Consider now a point in the shadow at distance r from 

 the centre, and at such a distance from the surface that 

 kr — z is of higher order than A Then 



kr — m = kr — z + iz^/3 



is also of higher order than z^ y and its real part is positive. 

 Accordingly, selecting the appropriate asymptotic expan- 

 sions, 



(f>nr= (Jcr 2 — m 2 ) 2 + m sin -1 m\kr — \mir-\-\Tr. ) 



Thus, with m = z — LZ~s(3, the significant values become 



R nr =krl(Pr 2 -z^f^(l-a 2 /r 2 y h ) . . (183) 



or unity at a sufficient distance. Moreover, neglecting 

 relative order z' 1 . 



so that the imaginary part of <j> nr may be written in the form 



on reduction. 



