Electric Waves round the Earth. 759 



of the oscillator from the centre of the earth of radius a are 

 respectively r and r t . If 2ir/k be the wave-length, z = kr, 

 z =ka, z 1 = kr i , and <f> and fa are similar functions of z and 



z x involving m or n+ -. The forms taken by these functions 



are determined by the relative values of m and their argu- 

 ments. Prof. Macdonald then writes this series as the 

 difference of two others, whose summations range from 

 1 to co and from 1 to z respectively. These series have the 

 same zero point, defined by 



+ d<£/drc + a<fc/an = O (2) 



and their principal parts arise from the terms on both sides 

 c£, but close to the term for which this is true. 



h or simplicity, we may restrict attention to the case in 

 which <f> and fa are identical, so that the transmitter and 

 receiver are equidistant from the centre of the earth, for this 

 is the case in which a tabulation of the final formula is given, 

 ^ow if n and z are not nearly equal, 



~d(f>fdtt = a — ^7r (3) 



yvhere sin «=(w+i)/s, and the zero point for the case <l> = fa 

 is given by 0=7r-2a, as in Prof. Macdonald's paper. 



Now let n 1 denote the value of n at this zero point. By 

 tie use of this value, which in the special case (f> = fa is given 

 by m + ± = z cos ±0, Prof. Macdonald obtains the principal 

 parts of the two component series, so that by subtraction, 

 k 4 3 is proportional to the integral 



- "J J*' ; • • (i) 



where the argument of the exponential in the integrand is 

 derived from 



'd' 2 (f>l'd'i 2 = sec a. (5) 



which is true when n 2 and z are not too nearly equal, if also 

 6 is not too small. But this condition has already been 

 introduced with the use of an asymptotic formula for a zonal 

 harmonic. This integral is finally reduced to a dependence 

 on FresnePs integrals in the form " 



J" 



dye-** 1 ** (6) 



where 



rj =2(a-r cos ±6)(\r sin \0) -* . . . (7) 



for the case of an oscillator close to the earth, X being the 

 wave-lenoth. 



