760 On the Bending of Electric Waves round the Earth, 



Now the objection to be urged is that this formula 

 cannot be used for points near the earth's surface. For 

 other points not very close, it does lead to the true sum 

 for the series S 43 , and, as Prof. Macdonald remarks, is in 

 agreement with a previous result. But when the oscillator 

 and receiver are both close to the surface, z and z 1 are both 

 practically equal to z , and consequently <f> and <j>i equal to 

 <j> for any value of n. The original series (2) commences 

 with the harmonic of order z for which <£ , now the common 

 value of the functions </>, is not of high order and in fact is 

 practically equal to ^ir. At no point in the series is <f) of a 

 form in accordance with (3). Now Prof. Macdonald has 

 replaced this series by two others each commencing with the 

 harmonic of order unity. In each of these, <f> can take a form 

 consistent with (3) for part of the range, and they both lead 

 to the same zero point. But it is not justifiable to take, as 

 the sum of the original series, the difference in the principal 

 values of these two series. For in determining a principal 

 value, it is definitely supposed that there are a large number 

 of harmonics on each side of the zero point for which <£, </> L 

 (and, moreover, R and Pi and similar functions) do not 

 change in type, and the changes of type, for harmonics be- 

 yond these, are ignored. But in the present case, it is 

 harmonics of different type which alone are concerned, as we 

 have seen, since the original sum extended from z to oo . 1\ 

 cannot therefore be lawful to equate their sum to the difference 

 of two series (from 1 to z , and from 1 through z to go ) 

 deduced by a method which assumes, even in the second of 

 these series, that the harmonics continue to co of the same 

 type, a type not found in the original series. 



These considerations appear to show definitely that the 

 formula given by Prof. Macdonald is not valid when the 

 transmitter and receiver are near the earth. Moreover, it 

 will be seen that when cj> and ^> x are not nearly equal to <£ , a 

 large number of harmonics of the proper type are found in the 

 original series (1), and the formula thus becomes valid. This 

 explains why, for points not close to the shadow, it is in accord 

 with previous results. 



We must conclude finally that the tables given for the 

 intensity produced by diffraction round the surface ought to 

 exhibit a very much smaller effect. In a later section of the 

 paper now being published " On the bending of electric waves 

 round a large sphere," tables of the exponential formula will 

 be given. In the form in which M. Poincare has left it, 

 there are two undetermined magnitudes. 



