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



power of* ka Trill appear in the writer's second paper " On 

 the bending o£ electric waves round a large sphere/' 



But very different results have been obtained by Prof. H. 

 M. Macdonald in a paper just published *, and the main 

 purpose of this note is an examination of Prof. Macdonald's 

 proof, in which there appears to be a flaw which gives rise 

 to the whole difference in the results. The problem is 

 treated first in its general case, in which the oscillator is not 

 close to the surface, and the effect at any point whatever is 

 investigated to a first approximation. In an appendix, the 

 main formulae of pure mathematics required for the solution 

 are given. These relate, in the first place, to the summation 

 of a type of oscillating series, and in this section the 

 summation follows the lines of that given by the writer 

 previously f , and particular cases of the general formula are 

 developed. 



The remainder of the appendix is devoted to an exami- 

 nation of asymptotic values of the Bessel functions, in 

 which results are obtained in accord with those of the papers 

 mentioned in previous notes. 



In the treatment of the actual physical problem, Prof. 

 Macdonald first shows that the principal value of the sum of 

 the harmonic series at all points not very close to the geo- 

 metrical shadow and outside it, is identical with that which 

 would be obtained by the methods of geometrical optics, 

 provided that certain terms are neglected which, as is stated 

 in a footnote, are only important in the neighbourhood of the 

 shadow. But when he proceeds to an examination of the 

 shadow, the method employed is liable to a criticism which 

 appears to be very decisive. In accordance with the usage 

 of the writer's first paper on the subject J, a point at which 

 the derivate of an exponent in an oscillating harmonic 

 series can vanish will be called a " zero point." Prof. Mac- 

 donald reduces the problem in this case to a determination 

 of the principal part of a series of the form, where A is the 

 same for every term, 



S 43 =A2; o 7n f (RR 1 )^~^ + ^ +m0_i7r) • • • (1) 



m being n + ~. The functions R and R^ do not oscillate 



in any part of the range of summation. The orientation of 

 the point considered is 0, and the distances of this point and 



* Phil. Trans. A. vol. ccx. pp. 113-144. 

 f Messenger' of Math., Oct. 1907. 

 X Phil. Mag. April 1910. 



