ELECTKIC WAVES OVER THE SURFACE OF THE EARTH. 121 



amplitude curve (II. ), are shown in fig. 2, in which the ordinates that separate the 

 half-bays are also drawn. The greatest ordinates are in the bay between the values 

 4 and 18 of n -8000 ; with this were taken the two half-bays from 4 to 3 and from 

 19 to 26. The next similar portion on the left consists of the half-bay from 34 to 

 27, the bay from 26 to 12, and the half-bay from 11 to 4; and the next 

 similar piece on the right consists of the half-bay from 26 to 33, the bay from 34 to 

 48, and the half-bay from 49 to 56. The ordinates common to two consecutive half- 

 bays were halved, and half taken as contributed by each. The sums of the terms 

 thus contributed to the series by the three pieces of the curve described above were 

 approximately 61, 14, and 5, which yield as sum 42. It may also be seen that 

 this is very nearly the sum of the whole series. After the critical middle piece 

 ( 4 to 26) the series may be broken up into sums of terms contributed by 

 pairs of consecutive half- bays, e.g., 26 to 33 and 34 to 41, are such a pair, 41 to 

 48 and 49 to 56 are the next pair, arid so on. The sums contributed by such 

 pairs have alternate signs and diminish continually in absolute magnitude. It 

 thus becomes easy to estimate the error made in breaking off the sum at 

 specified maxima, or minima, of the sine or cosine involved. In the particular 

 example of S ]2 for 9= 12, the error made in breaking ofF the sum at 34 and 56 

 was estimated in this way as about 1 ; and the value of S 12 for 6 12 was taken 

 to be about 41. 



15. The value of H expressed by (42) involves eight of these series, but the labour 

 of calculation can be reduced very much by observing the forms of the two expressions 

 in square brackets in (42). The terms written in the first lines of these expressions 

 represent waves travelling outwards from the originating doublet, with a velocity C 

 along the arc of a great circle drawn from the originating doublet to the point of 

 observation, the wave having at each point an amplitude and phase depending upon 

 the values of S n , ... at the point. The second lines represent a return wave. Now it 

 is evident that for moderate values of 6, say less than \-K, the amplitude of the 

 return wave must be very small, and thus we are led to expect that, at least 

 approximately, 



Su = S 12 , S 21 = S 22) S' n = S' ]2 , S' 21 = S' M . . . . (50) 



Aii analytical proof that the return wave is negligible in the case of perfect conduc- 

 tion has been given by MACDONALD( T ), arid the above approximate relations have been 

 verified numerically in a few cases. By assuming them throughout the labour of 

 calculation could have been considerably reduced. 



16. The values computed for S 12 , S 32 , S'u, S' 22 for the selected values of 6 are 

 recorded in the following table : 



VOL. CCXV. A. B 



