43() Dr. J. W. Nicholson on the Bending of 



noticed these points, and has published, in the current 

 number of the Rendiconti del Circolo Matematlco dl 

 Palermo*, an exhaustive investigation of the problem 

 amended in each of these respects. The general conclusions 

 to which he is now led are substantially in agreement with 

 those of the writer's paper, which were expressed in the 

 previous note. Instead of, at any point of the earth's sur- 

 face, an intensity due to diffraction of magnitude sufficiently 

 large to furnish an explanation of the observed phenomena, . 

 M. Poincare now finds an intensity containing an exponential 

 factor of large negative argument proportional to rr&d 9 where 

 is the orientation of the point from the oscillator, and m is 

 a magnitude roughly of order 10 6 , depending on the ratio of 

 the wave-length of the oscillation to the diameter of the 

 earth. 



But this agreement is only general in character, and does 

 not extend to the actual quantitative results. The object of 

 the present note is to point out that M. Poincare's expo- 

 nential formula does not appear to be of such general appli- 

 cation as his investigation would indicate. From my own 

 investigation, which proceeds on different lines to that of 

 M. Poincare at many stages, the exponential factor may be 

 obtained readily for points whose orientation, measured from 

 the oscillator, is not great. But for points at a greater dis- 

 tance, such that the argument of the exponential exceeds a 

 certain moderately large value, it ceases to be the important 

 term in the diffracted effect. The rate of decay of the 

 intensity round the surface, as given by M. Poincare's 

 formula, would therefore be much too rapid, although, as 

 stated in the previous note, it is certainly sufficiently rapid 

 to eliminate diffraction as a possible explanation of the 

 phenomena. 



M. Poincare's mode of solution is briefly as follows. The 

 electric force at any point on the surface of the earth is first 

 determined in the form of an harmonic series, the coefficient 

 of the zonal harmonic of order n being a meromorphic 

 function of n + ^ derived from the Bessel functions of this 

 order. This series ceases to converge when the oscillator is 

 placed on the earth's surface, thus necessitating some trouble- 

 some mathematical processes for its evaluation in this limiting 

 case. In the writer's paper, the magnetic vector has been 

 dealt with instead, chiefly on account of the fact that, by its 

 use, a convergent series may be obtained. M. Poincare then 

 proceeds to an examination of the infinities of the mero- 

 morphic function, with a view to its expression in a more 

 * Marzo-Aprile, 1910, pp. 169-261. 



