Elastic Solid ^Ether. 



555 



the medium and % be its arbitrary displacement (intrinsic + 

 additionally disturbed) ; let Xo be tne strain Unity (that h 

 linear vector function o£ a vector) and \jr the pure strain 

 linity, so that with <o for an arbitrary vector, 



Xo&>= - So>V • Vm ^0^= -i^oiS^Vi — iVi Sw7 7oi- (?) 

 The cubic rotational term is 



= - JCS V1V2V3 S^^o2^o3 - iCS ( V VfoWo (VWo) • 



This is easily proved by expressing % in the form 

 yfr + Y\ ( ), and verifying that the terms of Sf,? 2 ?3 

 S^o?i^o52^o?3 which are linear and which are cubic in \ 

 are identically zero. 



The volume integral of SV^Vs 87701^02^03 for the whole 

 of space reduces to a surface integral at infinity and is zero 

 when 7) converges properly. Putting it zero the cubic 

 rotational part of the potential energy is the volume integral 

 for all space of 



-|CS(VV)„)fo(VV"o)- 



Since this is zero when the curl is zero, it follows that the 

 cubic term gives rise to zero elastic resistance to irrotational 

 strain. Since the intrinsic strain is irrotational, and the 

 disturbed strain is small compared with the intrinsic strain ; 

 by neglecting all but the lowest order terms in the last ex- 

 pression we may therein suppose yfr to stand for the constant 

 (in time) intrinsic strain and Vy*7 for the superposed curl, 

 so that our original cubic term has taken a form which is 

 quadratic in the (disturbed) curl. 



Let p be the position vector of a point in the intrinsically- 

 strained position, v the disturbed displacement, ^ the dis- 

 turbed strain linity, ^ the disturbed pure-strain linity, all 

 reckoned from the intrinsically strained state. Remembering 

 the irrotational constant meaning of yjr in the last form 

 above of the cubic rotational term, put 



iCS|r (»=7ia>=— Sg>V • V$S 



(6) 



where g is a given scalar function of position whose first and 

 second space derivatives are everywhere finite, and therefore 

 h is a given self-conjugate linity. 



