252 Mr. H. M. Macdonald. The Bending of [Jan. 21, 



which is of some importance on account of its application to the propa- 

 gation of electric waves along the surface of the earth. The case is 

 that of a Hertzian oscillator placed outside a perfectly conducting 

 sphere. Let the radius of the sphere be a, and let the oscillator be at 

 the point C, whose distance OC from the centre of the sphere is r\ t 

 the direction of the axis of the oscillator being along the line OC. 

 The lines of magnetic force are circles, whose centres lie on OC, and 

 whose planes are perpendicular to OC. If y denotes the magnetic force 

 at any point P whose distance from OC is p, yp satisfies the differential 

 equation 



|~2 ( yp } ~ ~ J~ ( yp "> + §3 ( yp } + K<2yp = °' 



where z is the distance of the point P from some plane of reference 

 perpendicular to OC, and 27t/k is the wave-length of the oscillations. 

 Transforming to polar co-ordinates (r, 0), where r = OP and is the 

 angle COP, this equation becomes 



fl (yp) + — 1 1 , (yp) + *yi> -0 (I), 



in which fx = cos 0. The general solution of this equation, which is 

 applicable to the space external to a sphere, is 



yp = ^!{AnJ n+ | («-) -f (*)} (1 - ^§5^ , 



in which J m (kt) denotes Bessel's function of order m and V n (p) the 

 zonal harmonic of integral order n. It is, therefore, first necessary to 

 express the magnetic force due to the oscillator in this form. If y\ is 

 the magnetic force at the point P due to the oscillator, yi is the real 



part of C~ e l * (R V) , in which R is the distance CP and V is the 



dp R 

 velocity of radiation.* Writing 



Op ±\, 



and remembering that R 2 = r 2 + - 2rrifi, 

 this is equivalent to 



dp 

 when ?•<?*!, where 



mm 



K m (cKr) = [J_. m (/cr) - <*™J m (KT)].i 



2smw7r L 



* Hertz, ' Electric Waves,' Eng. Trans., p. 141. 

 f Macdonald, ' Proc Lond. Math. Soc.,' vol. 32. 



