( 404 ) 



^ 3. Before coming to the problem we have in view, it is necessary 

 to treat some preiiminarj questions. In the first place, we shall 

 examine the vibrations that are set up in an unlimited liomogeneous 

 and isotropic body subjected to given electromotive and magnetomotive 

 forces, changing with the frecpiency n. This problem is best treated 

 by using the complex vectors. 



We may deduce from (1) 



1 



rot rot S^ zzz — rot ^, 

 c 



or 



grad die .f) — A J^ r= - rot S (11) 



c 



and similarly from (2) 



grad die ^ — A (J = rot^ (12) 



c 



Again, always using the equations (1), (2), (9) and (10), we find 



dlv 53 = , dlv (£ = 0, 



div '^ = — div J^e, dio ^ ■=. — div ^g, 



rot (i = - ( rot (I- -f rot ^'e ) = ^ + - rot ^^ 



p 'pc p 



pqc p 



1 . . 1 .. 1 . 



rot $ = - {rot Sp + rot SDe) = ~^ -\ rot .'r^e 



q qc q 



1 . . 1 . 



— (^£-^ i^^)J^^rotS^e, 



pqc q 



SO that (11) and (12) become 



1 . 1.1 



L Jp S^ = — grad div .% -\ — .^e rot ^ci 



pqc pqc' pc 



^ 1 . 1.1. 



AS e- = — grad dlv ^^ -| ~ ^e -\ rot S^e- 



pqc pqc^ qc 



Tlie solution of these equations may be put in a convenient form 

 by means of two auxiliary vectors Ot and 0. If these are determined 



by 



A '31 ~i\ = ~(ie, (13) 



pqc 



A£l ^^a = -^e, ...... (14) 



pqc 



we shall have 



S:i ^:^ grad div Cl — ^ ^ b, -\ roi3l, . . . . (15) 



P>qc'^ ' pc 



