758 Prof. P. J. Daniell on Rotation of 



These are the same as the ordinary equations obtained for 

 a body with the same elastic constants X, /*, with density 



s instead of p or -=-, and acted on by a surface force 



c ' c 



u 2 

 normal to the surface = X — 2 pressing inwards. The solutions 



in any particular problems can be found by the ordinary 

 methods, if these two changes are made. 



For the general case Betti's Reciprocation Theory * can 

 be used. We shall denote by e xx the average value, not of 



e xx as defined above in terms of a, but of an e xz = —^ defined 



Ox 



in terms of the actual deformation. There will be apparent 



u 2 

 volume forces (M +>«.—/*) grad ^-g and apparent normal 



u 2 

 surface forces X 5-= inwards. 



2c z 



Let E denote Young's modulus, a Poisson's ratio, V the 

 volume. 



--ffl^Uij-^iiS) 



ET 



\ rr y 2 



where ^S denotes an element of surface., 



I, m, n are the direction cosines of the outward drawn normal. 



But JJ' x oTx^ 2 ) dxdydz= jjh xlds ~ \])^ dxd y dz 



Then 



M-fiCCi\ d (v?\ d (u 2 \ 



€ - = EV JJj t*di\p)-W%\&) 



-* z k(£$ dxd y dz 



~ ev JJJ 5? (1 " 2<7) dx dy dz ' 



* Love, ' Elasticity ' (2nd ed.), p. 170. 



