Elastic Bodies and the Principle of Relativity. 759 

 Let # = 0, ?/ = 0, 2 = be principal planes of the body 



and let K 2 = ^ \\\ x * & x ty & z -> e ^ c - 



Further, let Qz be the axis of rotation and suppose the 

 body to be symmetrical about this axis. 



Then w z =0, % = 0, co z = 0) -> 



k x —Ky —2^1 • 



Then 



L c 2 Ei 2c 2 _ 



— Newtonian mechanics would give the first term only, and 

 the average e xx , e are less according to the Principle of 

 Relativity. 



oy 2 h 2 



^ = 2^[-MM-^)-Hi-~^)] 



27 2 M °" 



This is the same as in Newtonian mechanics. 

 Similarly, using Betti's theorem it can be shown that 



e ? = e, x = e x = 0, as in Newtonian mechanics. 



We can define a " semi-rigid " body as one in which E is 

 infinite, not only as compared with p, but with pc 2 or M. 



Then e^=0, ^=^=^=0, 



*xx — @vri — 



4c 2 



Thus a body with infinite elastic constants, or infinitesimal 

 density, can rotate, but it does not remain " rigid " in the 

 absolute sense. 



div a = div b -j — ^— 2 



. - - , *> 2 &i 2 



rM(l-cr) M<q 



- w Ll L c 2 E c 2 EJ 



M(l-2<r) 



- C0kl ?W~' 



