SOO Proceedings of the Royal Irish Academy. 



while for the stram components a, h, c,f, g, h, we have 



P 



4:7TfJl 

 P 



« = (1 + 2£) 



& = (1 + 2e) 



ccdV ^ 



-3 -,— dq - o£ 

 r^ dx 



P 



r^ dy 

 z dV 



dq - OE 

 dq - Se 



1 dV 



- cos-A -^— dq 

 r- ar 



1 r^r , 



- cos-u — - dq + E 



dq~-- dq, 

 r- clr 



■ IdV 

 )■• clr 



r' dz 

 when A, /./, v are polar angles of r, 



1 . ; f[f 1 dV 



— cos-y rta + £ — -7- '^^'7, 

 r- )■' dr 



OTTjil 

 P 



Stt/j. 



p 

 p 



g = (1+ 2e) 



h = (1 +2 £) 



dV dV\ 



dz dy 



dqf dV dV\ , 



dq cos u cos )' -r- , 



1 / dV clV\ ^ ^ 



r" \ dy dx j 



The dilatation A will be given by 



dV 

 dq cos 1' cos A -^-, 

 dr 



7 ^ '^^ 



dq cos A cos u -^- . 



This will vanish for 



. = a..)Jj|4:.i.,. 



1 + 2£ = 0, i.e. 



A 



-^^t 



1; 



i.e., for an incompressible medium, the expressions for the components of 

 strain undergoing corresponding simplification. 



Suppose now the matter electrical and the distribution superficial, and we 

 have the case of a dielectric field bounded by conductors. We may represent 

 the distribution as the limit of a spatial of uniform electric density and 

 varying thickness dn of layer. The surface unital charge m is then 

 connected with the solid p by 



q)uW = pdS dy or dM = —. 



P 

 Remembering now that the electric force is normal to the surface of the 



conductor, and denoting by cos A, cos B, cos C, the direction cosines of normal, 



we find the following expressions for displacements and strains : — 



hu = (1 + e) - nv cos AdS - £ 



iii'x du), 



hv ==(! + £) - lid cos BdS - £ m-q dw. 



kio = (1 + e) 



nd cos CdS - £ 



iidz dh)^ 



