ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 119 



It seems to be unnecessary to print a complete table of the results. Six figures 

 were retained in computing u n , v n , R, n , and this made it possible to compute tan x to 

 four figures, as above. The computation was carried from q a = ^- to q n = ^Mp. 



The corresponding values of 



Z - . R a cos 2 Xn , (II.) && ^ z - . E B cos 2 Xn tan Xn , 



(HI.) t"R J*. . K,, cos< Xa (1 -tan' Xn ), (IV.) (2& J *- . R n 2 cos 1 Xn 2 tan Xn) 



2- ' it/ Z ib 



were then computed for the same values of q n , four figures being retained. The 

 general character of the results can be exhibited to the eye by marking on squared 

 paper the points which have the above expressions as ordinates and the corresponding 

 values of n z, or n 8000, as abscissae, and drawing smooth curves through the 

 points. The four curves, numbered I., II., III., IV., in the same order as the four 

 expressions, are shown in fig. 1, p. 120. In order to keep the figure within bounds, 

 the values of the expressions III. and IV. have been ' divided by 1 before marking 

 the points representing them. 



14. The next thing to be done was to sum the series denoted by S n , ... for a number 

 of values of 0. The values chosen were Tr/30, ir/20, v/15, Tr/12, Tr/10 or 6, 9, 12, 

 15, 18. Every term of each of these series is the product of a number shown by an 

 ordinate of one of the curves I., ... IV., the ordinates of the curves III., IV. being 

 multiplied by 10, and a simple harmonic function of n, actually a sine or cosine of 



or 



Thus the terms of the series can be represented graphically by ordinates 

 corresponding to integral values of n 8000 as abscissae, and a smooth curve drawn 

 through the extremities of the ordinates. The curve fluctuates like a curve of sines, 

 but the amplitude is variable, and the terms may be divided into groups of one sign 

 by means of the points in which the curve cuts the axis of abscissae, and these again 

 into halved groups by means of the abscissae for which the corresponding sine or 

 cosine is a maximum or a minimum. The portions of the curves that correspond to 

 these groups and halved groups of terms may be described as " bays " and 

 " half- bays." The method adopted for summing the series is analogous to the method 

 employed by FRESNEL in problems of diffraction, wherein use was made of the notion 

 of " half-period elements " ; when the amplitude is nearly constant the terms 

 belonging to a bay are nearly cancelled by those belonging to the two neighbouring 

 half-bays. 



It seems best to illustrate the method by explaining how S 12 was evaluated for 

 & = 12. A portion of the fluctuating curve, and the corresponding portion of the 



