and Electric Waves upon Small Obstacles. 20 



and at distances outwards which are infinitely great in com- 

 parison with the wave-length (A,) . The method proceeds by 

 considering in the first instance what occurs in an inter- 

 mediate region, where the distance (r) is at once great in 

 comparison with the dimensions of the obstacle and small in 

 comparison with X. Throughout this region and within it 

 the calculation proceeds as if A were infinite, and depends only 

 upon the properties of the common potential. When this 

 problem is solved, extension is made without much difficulty 

 to the exterior region where r is great in comparison with X, 

 and where the common potential no longer avails. 



At the close of the paper a problem of some importance is 

 considered relative to the escape of electric waves through 

 small circular apertures in metallic screens. The case of 

 narrow elongated slits has already been treated *. 



Obstacle in a Uniform Field. 



The analytical problem with which we commence is the 



same whether the flow be thermal, electric, or magnetic, the 



obstacle differing from the surrounding medium in Conduc- 

 ts m C3 



tivity, specific inductive capacity, or permeability respectively. 

 If </> denote its potential, the uniform field is defined by 



cfr — ux + vy + wz; (1) 



n, v, w being the fluxes in the direction of fixed arbitrarily 

 chosen rectangular axes. If i/r be the potential in the uni- 

 form medium due to the obstacle, so that the complete poten- 

 tial is 4> + y}r, yjr may be expanded in the series of spherical 

 harmonics 



*=* + | + V"-« .... (2) 



the origin of r being within the obstacle. Since there is no 

 source, S vanishes. Further, at a great distance S 2 , S 3 , . . . 

 may be neglected, so that yjr there reduces to 



^ = B_, = AWBW m 



The disturbance (3) corresponds to (1). If we express 

 separately the parts corresponding to u, v, w, writing 

 A'=A 1 u + A 2 v + A z w, &c, we have 



r d y{r = u {A x x + B^ + C x z) 



+ v{A 2 x + B 2 y + C 2 z) 



+ w{A 3 a; + B 3 y + Csz)', (4) 



* Phil. Mag. vol. xliii. p. 272. 



