and EUctric Waves upon Small Obstacles. 43 



An important particular case is obtained by making 

 H / = yo, /jl' = 0, in such a way that V x remains finite. This 

 is equivalent to endowing- the obstacle with the character of 

 a perfect conductor, and we get 



°=-™*-*-(0 $+?.}, ■ ■ (88) 



which, when realized, coincides with (72). 



The other two-dimensional electrical problem is that in 

 which everything is expressed by means of R, the electro- 

 motive intensity parallel to z. The conditions at the surface 

 are now the continuity of II and of fi~ l dR/dn. Thus K and 

 /j, are simply interchanged, /u, replacing a and K replacing 

 1/m in {Q6), (70), cf> and yjr also being replaced by R. In the 

 case of the circular cylinder 



T> _ _ M a « g -ar fjL^f I K ~ R/ + P'-* 1 X I (< RON 



R- kae [ 2lkr ){ 2K + fl < +fl -p (8J) 



corresponding to the primary waves 



R = e ikx (90) 



If in order to obtain the solution for a perfectly con ducti no- 

 obstacle we make K7 = c© , /// = 0, (89) becomes infinite, and 

 must be replaced by the analogue of (83). Thus for the 

 perfectly conducting circular obstacle 



j. _ e~ ikr / it \i 



~y + Log{iika)\2ikr)' ' ' ' " ^ 

 which may be realized a? in (84). 



The problem of a conducting cylinder is treated by Prof. 

 J. J. Thomson in his valuable ' Recent Researches in Elec- 

 tricity and Magnetism/ § 364 ; but his result differs from (84) , 

 not only in respect to the sign of l\ but also in the value of 

 the denominator *. The values here given are those which 

 follow from the equations (9), (17) of § 343 'Theory of 

 Sound.' 



Electric Waves in Three Dimensions. 



In the problems which arise under this head the simple 

 acoustical analogue no longer suffices, and we must appeal to 

 the general electrical equations of Maxwell. The components 

 of electric polarization (/, g, h) and of magnetic force (a, /3, 7), 



* It should be borne in mind that y here is the same as Prof. Thom- 

 son's log y. 



