116 PROF. A. E. H. LOVE ON THE TRANSMISSION OF 



are those for which n does not differ too much from z, and that, in summing these, the 

 factors yPvo/v/ftn and e l( *""*"' o) may be omitted. 



10. For the series, which represents the effect of curvature without resistance, 

 methods of summation have been devised by MACDONALD( ; ), PoiNCARE^ 8 ), and 

 NiOHOLSON( 9 ) ; and still another method has been devised by MACDONALD( U ). All 

 these methods depend upon a transformation of the series into a definite integral, and 

 an approximate evaluation of the integral. POINCARE( S ) did not press his method so 

 far as to tabulate numerical results, but concluded that the expression for the electric 

 force normal to the surface, at an angular distance 6 from the originating doublet, should 

 contain a factor of the form e~ A *. NIGHOLSON( ! ') went further in the same direction, 

 obtained a formula for the magnetic force containing such an exponential as a factor, 

 and deduced definite numerical results. MACDONALD( T ) also obtained definite numerical 

 results which cannot be reconciled with those of NiCHOLSON( 9 ). The discrepancy 

 was discussed by NICHOLSON( I(I ), who traced it to an alleged flaw in the analysis used 

 by MACDONALD( T ), and it was discussed also by MACDONALD( U ), who pointed out a 

 difficulty in the analysis used by POINCARE( S ) and NICHOLSON^). Fresh numerical 

 results were deduced by MACDONAL,r>( n ) from a new method of summing the series^ 

 but they do not agree with those found by NiCHOLSON( 9 ), or with those previously 

 found by MACDONALD( T ) himself. 



11. It appears to be desirable to devise a new method of summing the series, to 

 apply it to obtain definite numerical results in regard to the problem of the perfect 

 conductor, and to compare these with the results obtained by MACDONALD and 

 NICHOLSON, further to apply it also to calculate the effect of resistance, and finally to 

 compare the results with those which have been found by experiment. The method 

 which I have used is to compute a sufficient number of terms of the series and add 

 them together. It appeared to be best, in the first instance, to do the work for a 

 particular wave-length, and even with this limitation considerable labour was 

 required. The quantity chosen for calculation was the magnetic force H at a point 

 on the bounding surface. 



According to (15), (37), and (38), and the results already cited as known in regard 

 to the behaviour of the functions R n and x* , we have 



H = 2 ' R 



where the summation now refers to the relevant terms, and 



-z. .......... (40) 



