and Electric Waves upon Small Obstacles. 35 



IS is to be determined by the condition that when r = 



4:TTr 2 df/dr = k 2 T, 

 so that S = -FT/4tt, and 



FT e~ ikr ttT e~ 



■ikr 



r 4?r r X- r v y 



This result corresponds with wi'— » representing absolute 

 incompressibility. The effect of finite compressibility, differing 

 from that of the surrounding medium, is readily inferred by 

 means of the pressure relation (Sp=ms). The effect of the 

 variation of compressibility at the obstacle is to increase the 

 rate of introduction of fluid into T from what it would 

 otherwise be in the ratio m : m' ; and thus (45) now becomes 



7rT in' — m e~ ikr ,.„,. 



f=—^> — < r> • • • • (46) 



' X~ in r ' 



or if we restore the factor e ikYt and throw away the imaginary 



part of the solution, 



7rT m' — m 7 , TT N ,._. 



^=-^7 2 j- cos k{Vt-r). . . (47) 



This is superposed upon the primary waves 



(]> = cosk(Vt + x). ... . . . (48) 



When m' = 0, i. e., when the material composing the 

 obstacle offers no resistance to compression, (47) fails. In 

 this case the condition to be satisfied at the surface of T is the 

 evanescence of 8p, or of the total potential (^> + ^). In the 

 neighbourhood of the obstacle 0=1; and thus if M' denote 

 the electrical " capacity " of a conducting body of form T 

 situated in the open, i/r= —W/r, r being supposed to be large 

 in comparison with the linear dimension of T but small in 

 comparison with X. The latter restriction is removed by the 

 insertion of the factor e~ ikr , and thus, in place of (46), we now 

 have 



HI 



-ikr 



yjr= — (49) 



The value of M' may be expressed when T is in the form of 

 an ellipsoid. For a sphere of radius Ti, 



M^R; (50) 



for a circular plate of radius R, 



M / = 2E/tt (51) 



D2 



