Electric Waves round the Earth. 217 



radius of the sphere, and k\=2ir, the argument of the Bessel 

 functions is z=ka. This series can be shown to be formally 

 identical with a particular case of the general series derived 

 by the writer in the investigation mentioned above, without 

 the use of Fred holm's integral as an intermediary. The 

 particular case is that in which the electric effect is only 

 required on the surface of the sphere. 



In the summation of this series lies the error of M. Poincare/ s 

 treatment. Noting that when the order and argument (m 

 and z) of a Bessel function are large, and z>m, the asym- 

 ptotic expansion contains a factor (z 2 — m 2 )~*, and that if 

 z<m, it contains (m 2 — -c 2 ) - *", M. Poincare has assumed a 

 continuity between these expansions when m = r. Thus 

 J m (m), when m is large, becomes a very large quantity, and 

 this is the ultimate basis of the conclusion reached, that 

 diffraction can explain the phenomenon presented, for the 

 terms near m=z in the series are treated as the most im- 

 portant. Now in a series of papers published by the writer 

 in the Phil. Mag.* the complete scheme of asymptotic 

 expansions of the Bessel functions of real argument, for all 

 ranges of the ratio of large argument to large orderf has been 

 developed. This scheme was originally obtained solely on 

 account of its necessity for the present physical problem. It 

 indicates that the function J m (m) is not large, but very small, 

 and that when m and z are nearly equal, J m (~) is dependent 

 upon an Aiiy's integral. The factor (z 2 —m?)-l disappears 

 when z — m is small in comparison with :. In the final 

 paper f, the continuity of this sequence of expansion- i- 

 demonstrated. 



It follows that the sum of the harmonic series, in the phy- 

 sical problem, may be of a very different order of magnitude J, 

 and in the investigation of the writer this is the case. In the 

 general case, the summation of the series proceeds on different 

 lines according to the position of the point of space at which 

 the electric effect is desired. For points not close to the 

 region of geometrical shadow, the important harmonics arc 

 those containing Bessel functions whose order and argument 

 are not nearly equal, though of the same order of magnitude. 

 The first approximation to the final sum gives the effect of the 

 sphere regarded as a plane reflector. This is in accord with 

 physical needs. On the confines of the geometrical shadow. 



* Dec. 1007, Aug. IOCS, July 1909. Feb. 1010. 



t Phil. Mag. Feb. 1910. 



X During the passage of this note through the press, M. Darboux has 

 kindly referred me to a note communicated by M. Poincare more recently, 

 in -which this criticism is independently made. 



