756 Prof. P. J. Daniell on Rotation of 



The potential function depends only on the rest-defor- 

 mations a. It can be seen that 



u 2 

 (i.) div a=div b + -^2 ? 



(ii.) curl a = curl b, 

 (iii.) grad a* = grad ^ + (|^) • u , 



(iv.) at the surface if 8n denotes an element of normal 

 to the surface, and if n x is a unit vector in the direction of 

 the normal 



"da ~db , N u 



on on v ' 2c 2 



Only bodies symmetrical about their axes of rotation will 

 be considered, and then u is perpendicular to n^ 



Or Sa = ^.. 



Let us put 



oa x _ oa^ _ ~doz 



O* ' eyy ~^y ' 8zz ~oz 



_l(oa z daA _l/oa x . oa z \ lfoa„.oa x \ 



e »*-2W + oT> '"-2V57 + ^> ^ = 2<^ + ^> 



The ordinary potential function is given by W where 



2W' = (\+2/t)(g IC 4- % + £^) 2 + 4^(e 2 y . + <? 2 ^ + e 2 xy — e yy e zz — e zz e xx - e xx e yy ) . 



This can be reduced to the form 



2W = \(div a) 2 -/z(curl a) 2 + 2/*[(grad a*) 2 4- (grad %) 2 -f (grad a,) 8 ]. 



The " action " would be given by T— W, where T is the 

 kinetic energy, 



T= jP-'^gM^, ifM= P o 2 . 



We can bring this kinetic energy into the expression of 

 the potential function by assuming it to be 



W=-M-W / 



per unit 4-volume in (^ y %^ ) space. 



W= — i\(div a) 2 -\-ifi(cuY\ a) 2 — fi [(grad aj 2 + (grad %) 2 + (grad a*) 2 ] -M. 



X, fi are the ordinary elastic constants, M = c 2 X the 

 density p. 



* Love, ' Elasticity,' 2nd edition. 



