Electric Waves round the Earth. 437 



appropriate form by the use of Cauchy's theorem. He finds 

 that one infinity is of greater importance than the others, 

 and corresponds to a value of n, whose imaginary portion is 

 very great, and of order mi. The function is then replaced 

 by the term corresponding to this infinity in the ordinary 

 way. 



Certain formula? of summation of series of zonal harmonics 

 are then developed, in their first approximations only, and 

 finally, the particular series in question, with the coefficient 

 of the harmonic replaced by the formula correspon ding- 

 to its chief infinity. The result stated above is the con- 

 sequence. 



Now the terms neglected during the summation do not 

 necessarily all lead to exponentials, in their final contributions 

 to the result, in the same manner, and may therefore lead to 

 a sum which, for a large orientation from the oscillator, 

 does actually transcend that due to the terms retained. For 

 however small their sum may be, if its decrease with orien- 

 tation is not exponential, it may soon be of greater importance 

 than M. Poincare's exponential effect, as the point moves 

 round from the oscillator. This is undoubtedly the origin of 

 the discrepancy still existing between M. Poincare's results 

 and those of the writer, who has found, by a different process 

 of summation, that the effect is certainly exponential at first, 

 but only for a few hundred miles. As soon as this expo- 

 nential becomes smaller than a certain (not very well defined) 

 limit, its magnitude has really fallen below that of certain 

 other non-exponential portions of the sum, which although 

 decreasing as the orientation increases, do not decrease so 

 rapidly. These portions are determined to a great extent by 

 low order harmonics in the series. 



For the great distances mainly in question, therefore, 

 another formula would appear to be necessary, and this will 

 be found in the investigation appearing shortly. 



En conclusion, it may be pointed out that M. Poincare's 

 modified expressions for the Bessel functions whose argument 

 and order do not differ widely, are in accord with the values 

 deduced by the writer in earlier papers, and mentioned in 

 the previous note. 



Trinity College, Cambridge, 

 February 10, 1910. 



