[ 754 ] 



LXXXI. Rotation of Elastic Bodies and the Principle of 

 Relativity. By P. J. Daniell, Assistant-Professor of 

 Mathematics , Rice Institute, Houston, Tea;.* 



IT is well known that the rotation of a " rigid " body 

 about a stationary axis is inconsistent with the Principle 

 of Relativity. In fact, the circle along which any part moves 

 is contracted while the radius is unaltered. It has been 

 suggested that a rotating circular disk might buckle, but it 

 could then no longer be regarded as rigid. Herglotz, in a 

 paper on the mechanics of deformable bodies f, has shown 

 a method by which problems relating to elastic bodies may 

 be solved. The method depsnds on the variation of the 

 integral of a potential function of the rest-deformations; 

 that is to say, if each elementary part of the body is referred 

 to axes with respect to which it is at rest, the deformations 

 in terms of these new axes are the rest-deformations. Here 

 this method has been used to consider the question of the 

 rotation of an isotropic elastic body. It is found that even 

 when the elastic constants are infinite, rotation is possible,, 

 although the rest-deformations are not zero. The body is 

 not " rigid " in the absolute sense of Born's definition. It 

 might perhaps be called semi-rigid. By some the Principle 

 of Relativity has been extended so as to include the assump- 

 tion that even elastic waves cannot be propagated with a 

 velocity greater than that of light ; then the ratios of the 

 elastic constants to the density cannot exceed the square of 

 the velocity of light. The elastic ratios of all material 

 bodies, even of so-called " incompressible " fluids, do not 

 approach this value. But in electron theory the " semi- 

 rigid " body whose rotation is not inconsistent with the 

 Principle of Relativity may be even more useful a con- 

 ception than Born's rigid electron. 



Only small deformations will be considered, and the square 

 of the ratio of the velocity of any part to the velocity of light 

 will be taken as small. Squares and products of such 

 quantities will be neglected throughout. 



Let r denote a position vector in the undeformed body 

 referred to a system S ; 

 r + b denotes the corresponding deformed position vector 

 referred to space-axes rotating with the body ; 

 r' denotes the corresponding deformed position vector 

 referred to S. 



* Communicated by the Author. 



t Herglotz, " Mechanik des Deformierbaren Korpers," Annalen der 

 Physik, xxx vi. p. 500. 



