Elastic Bodies and the Principle of Relativity. 757 



The required differential equations are found by making 

 the variation 



8$$Wdxdydzdt = 0, 



or S jjjj Wdx dy dz(Kdt) = 0, 



or S^jjWKdxdydz = 



if the rotation is uniform. 



In the elastic terms of W, K can be put equal to 1 ; K is 

 retained only when it is multiplied by M, and then the 

 kinetic energy term appears. 



This variation is for all possible variations 8b if 



u 2 

 X grad div a + //, curl curl a + 2/ju (div grad) a + M grad ^ = 



throughout the volume, and if at the surface 



Xnj div a + fiCiii x curl a) + 2a^~- = 0. 



on 



These are exactly the equations which occur ini the 

 Newtonian mechanics {cf. Love, ' Elasticity ') ; but they 

 refer to a, the rest-deformation, not b, the actual defor- 

 mation. 



We must use the transformations (i. to iv.) above. Then 



A, grad div 6 -f^ curl curl b + 2/ju (div grad) b 



+ {\ + W) grad jj +2p (grad .«) ~ =0, 



(grad .u) u = (u . grad) u + u div u, 



div ?£ = div (o)X?«)=0, 

 (u . grad) 21= (cox u), 

 ■£ grad u 2 = (u . grad) u + (u X curl u) . 



But curl u = 2(o 



or (it X curl u)=—2 (o) xu) = —2 (it . grad) u, 

 so (grad . u)u= — ^ grad w 2 . 

 Hence the volume equation becomes 



(v.) (\ + 2/l&) grad div &— /a curl curl b 



+ ( X -^ + M)grad^=0, 



and the surface equation becomes 

 (vi.) Xwj div 6+/a(wiX curl 6) 



