256 



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



where e = o-k/47tV, and yip is the real part of /(/'), and o- is the specific 

 resistance of the material of the sphere, the other symbols having the 

 same meaning as before; it being assumed that e, which for ordinary 

 metallic conductors is of the order 10" 7 when the wave-length is about 

 10 cm., is small. If F' now denotes the electric force normal to the 

 surface, F differs in phase from F (the electric force normal to the 

 surface when the sphere is a perfect conductor) by a small amount, 

 and the ratio of the amplitude of F' to F is 1 - Je 2 . The effect of 

 imperfect conduction is therefore to diminish the electric force normal 

 to the surface, but only by an inappreciable amount when the obstacle 

 is as good a conductor as an ordinary metal ; for sea water, taking 

 <t = 10 10 , the correction is less than one part in a thousand.* 



4. The effect of a rigid spherical obstacle on the waves of sound 

 sent out from a source, when the wave-length is small compared with 

 the radius of the sphere, can be obtained by an analysis which s 

 almost identical with that given above. The result is that at any 

 point on the sphere at a distance from the source great compared with 

 the wave-length, 



where <f>i is the velocity potential at the point due to the source, and 

 <jf> is the actual velocity potential there. There is, therefore, no true 

 shadow near the surface of the sphere. Lord Eayleighf has discussed 

 the effect of a rigid sphere on the waves sent out from a source close 

 to the surface, and found that there was no indication of the forma- 

 tion of a shadow for wave-lengths greater than half the circumference. 

 The above statement completes the investigation. The conditions for 

 the formation of a distinct shadow, when waves of sound are incident 

 on an approximately rigid obstacle, follow • they are of the same type 

 as those already stated for an approximately perfectly conducting 

 body on which electric waves are incident. 



5. The results of § 2 have an immediate application to the question 

 of the propagation of electric waves around the surface of the earth. 

 Let C be a place on the earth's surface from which waves are being 

 sent out; these waves may be supposed to be due to an oscillator 

 placed vertically. The electric force acting on a receiver at a place, 

 whose angular distance from C measured along a great circle is 0, 

 will be F, given by equation (4) § 2, when the distance of the receiver 

 from the oscillator is great compared with the wave-length. In this 

 case f\ is nearly equal to a, and may be put equal to it for values 

 of for which equation (4) applies ; thus F = Fi (1 - sin It is 



* For the waYe-lengths actually used the correction is bss than one part in a 

 hundred millions. 



+ ' Theory of Sound,' vol. 2, § 328. 



