﻿of Elastic Stresses in an Isotropic Body. 281 



4. The inclusion of inertia terms. 



We must now modify equations (1) by writing pii, pv, 

 pib on the right-hand side, where u, v, to are to have the 

 values given in (3). 



We consequently replace P by P — p{9< 2 + 6 z — 6 x )\2n, 

 Q by Q-K& + 01-00 /2», R by R- /0 (g 1 + g a _g 3 )/2n, 

 S by S + /^h/», T by T+/njr 2 /w, U by U + p^/n. 



With these alterations the equations (2) still hold good. 



In forming our criteria, we equate values found from (2) 

 to values given by the stress-strain relations deduced 

 from (3). In these latter P, Q, R, S, T, U are to have 

 the original values of these quantities and not those which 

 replace them as above. 



Consequently, as our first step in the criteria, in place of 

 V 2 ^i = etc. we obtain 



nV 2 ^i = piri, »Wa = H^, «V 2 fs = P^ 

 and similarly 



™V 2 %1 = PXU n V 2 %2 = P%2, WV 3 'X8 = P%3- (8) 



In place of the last stage which gave the metric criterion, 

 we find 



, o ^ 2 %i j 2 d 2%2 ; o ^ \ 



oyoz dx^z d^ctyj J 



and finally, 



+ ^^m-°- (9) 



These are the modified form of (5) and (7). 



5". Inclusion of bodily forces, with particular reference 

 to gravity on the surface of the Earth, and to 

 centrifugal forces. 



In any case we shall have to consider the alteration made 

 in equations (1), and their solution in (2) by the introduction 



