and Electric Waves upon Small Obstacles. 39 



boundary, the constant S is to be found from the condition 

 that when r = Q 



2ttt dyjr/dr ~ k 2 T, 



T denoting now the area of cross-section. When r is small, 



dD {kr) _ 1 m 

 dr r' 



and thus S = £ 2 T/27r, 



+ = -sW=-sfe) r '- • (65) 



when r is very great. This corresponds to (45). 



In like manner, if the compressibility of the obstacle be 

 finite, 



k 2 T / 7r \§m'—m ., /ees 



T 7r \ZikrJ 2m 



The factor i~^=:e~^ a ; and thus if we restore the time-factor 

 e mt , and reject the imaginary part of the solution, we have 



2irTm' — wi 27r,„ , . ._ . 



^= rrr- -> / ■ cos -r-( Ve — r— AX). . . (b7) 



corresponding to the plane waves 



( j ) = C o S ~(Yt + x) (68) 



A. 



In considering the term of the first order we will limit 

 ourselves to the case of the cylinder of elliptic section, and 

 suppose that one of the principal axes of the ellipse is parallel 

 to the direction (x) of primary wave-propagation. Thus in 

 (34), which gives the value of ^ at a distance from the 

 cylinder which is great in comparison with a and b, but small 

 in comparison with X, we are to suppose u = ik, v = f at the 

 same time substituting a, a' for ///, (j, respectively. Thus for 

 the region in question 



ab.ikx (a' — a)(a + b) , . 



+ =~2? o>a + ab ; ' ' ' (69) 



and this is to be identified with SxD^r) when kr is small, 

 i. e. with Sj/Ar. Accordingly 



„ _as ik 2 ab {or' — o)(a + b) 

 1— r 2 a' a + ab ' 



