13 Q PROF. H. M. MACDONALD ON THE DIFFRACTION OF 



the above expression becomes 



f* 

 -2 -'''<* sin 2 a,<r" | e~ t/f ^c[rj, 



i(. 



where 



fy, = ('f r sin ) (21))' 2 (Xr/'j cos a cos i)~ 1/v; , 



hence 



In the corresponding part of S 3 has the opposite sign and therefore the sum is 

 negligible in comparison with S 43 . 



If p is the perpendicular from O on O,P, p = r sin a, and when p is greater than a 

 the point P is outside the boundary of the geometrical shadow, -q (i is negative and 



increases rapidly as /> increases, and the above value S 43 tends to the value found in 

 the case where the point P was supposed not close to the boundary. When p is less 

 than ft the point P is inside the boundary of the geometrical shadow, and the above 



value of S, r , is the principal part oft//, that is 



where 17,, has the value given above.* This result expresses the ratio of the magnetic 

 force at a point inside the boundary of the geometrical shadow to the magnetic force 

 due to the oscillator alone in terms of one quantity 77,,. For values of rj greater 

 than 5 integration by parts will give a sufficient approximation, for values of r) less 

 than 5 it is convenient to write the above in the form 



$ = -!/,, [1- (L + M) -i(L-M)], 

 where 



,-'ln filn 



L = I cos \Tnfd-r), M = sin \irrj' VZr?, 



* fl ^ 



and use GILBERT'S tables for these integrals. An important particular case is that 

 for which (), and P are both close to the surface of the sphere, as in the case of 

 wireless telegraphy. The tables given below show how the amplitude of the 

 oscillations diminishes as the distance along the earth's surface from the transmitter 

 is increased. In calculating these tables the transmitter and receiver have been 

 taken to be vertical antennae, the fundamental wave-length is five times the height, 

 and the results are given in two cases : for oscillations of wave-length one-fifth of a 

 mile corresponding to antennae of height 211 feet, and for oscillations of wave-length 

 one-quarter of a mile corresponding to antennae of height 264 feet. The first column 

 gives the angular distance of the receiver from the transmitter, the second column 



t The investigation assumes that the series in f obtained by the substitution n = mj + f converge for 

 values of n up to , and this will be so if P is near the boundary of the shadow, 



