Rotation of Elastic Bodies, 755 



If u' is the vector velocity of an element, 



dr'=dr + db+u'dt. 



Let 



dr = dr + da 



denote the differential o£ the corresponding position vector 

 obtained by making a Lorentz transformation to space-time 

 axes with regard to which the element is at rest. 



Then da is the rest-deformation differential vector. 



The Lorentz transformation gives 



dr =dr' + -^(V . dr') -ffu'dt, 

 where 



and c is the velocity of light, 



ecu 

 or dr = dr + db+ -j (u' . dr + db) . 



Instead of u! consider 



u= (coXr), 



where co is the rotation vector and (eo x r) is a vector 

 product. 



Then, neglecting squares of small quantities, 



dr = dr + db + — ? (u . dr), 



?/ 

 or da — db 4- 5-5 [u . dr) . 



Again, 



dt =-@ 2 (2i f .dr')+/3dt, 

 c 



== - 1 (u 1 . rfr + <ft) + £ (l - ~)<fc. 



If we consider t as changing while r, and therefore 6, is 

 fixed (i. e., uniform rotation), 



=:(!— -^jdt. or K<fa let us say. 



