and Electric Waves upon Small Obstacles. 33 



In the case of a very elongated ovoid L and M approximate 

 to the value 2ir, while N approximates to the form 



N=4^(logf-l), .... (32) 

 vanishing when e=l. 



In liuo Dimensions. 



The case of an elliptical cylinder in two dimensions may be 

 deduced from (12) by making c infinite, when the integration 

 is readily effected. We find 



L _4rt M= 4™ _ 



a + b a+b 



A and B are then given by (14) as before, and finally 



, ab(a + b) f (fi'—fi)ua! (//— fi)vy ~\ ru] 



^~ 2r 2 I /ui + fi'b + fib + fi'a J ' ' l ' 



corresponding to 



ty=.ux-\-vy (35) 



In the case of circular section L=-M = 27r, so that 



_aV-^ 



r v fi' ■+ fM v Jl v ' 



When b = 0, that is when the obstacle reduces itself to an 

 infinitely thin blade, yjr vanishes unless /// = or fi' = co . In 

 the first case 



(/*'=<>) *=^?; ■ • • • (37) 



in the second 



2r 3 ' 



(/ / = co) +=-~ (38) 



Aerial Waves. 



We may now proceed to investigate the disturbance of 

 plane aerial waves by obstacles whose largest diameter is 

 small in comparison with the wave-length (X) . The volume 

 occupied by the obstacle will be denoted by T ; as to its 

 shape we shall at first impose no restriction beyond the 

 exclusion of very special cases, such as would involve 

 resonance in spite of the small dimensions. The compressi- 

 bilities and densities of the medium and of the obstacle are 

 denoted by m, m'; a, a' ; so that if V, V be the velocities of 



Phil. Mag. S. 5. Vol. 44. No. 266. July 1897. D 



