418 



Sir William Thomson on the 



second approximation, correcting (1) and (2) in which only 

 forces in simple proportion to strains are taken into account, 

 we may now have, for the case of A = 0, 



W=iffidxdydz 

 f-^r/dw dv\ 2 (du div\ 2 /dv du\ 2 ~] . T \ /q X 



where I 4 denotes an essentially positive homogeneous function 



« .-, xl i P du du dv dw dw . 



ot the fourth degree of -=-> -j- f ... j-, . . . ^— > -7-, isotropic 



relatively to (a, y, z). 



8. The equations of motion of the general elastic solid, 

 taken direct from the equations of equilibrium*, are, in terms 

 of our previous notation adopted from Green, and with f to 

 denote density, 



d?u _ dP oTU dT 

 *dt 2 ~~ dx + dy + dz ' 



.d?v 



dt 2 



dJJ_ + dQ dS 

 dx dy dz 



W, 



t #w = dT dS dU 



dt 2 dx dy dz 



where u, v, w denote (as above from Green) displacements, 

 not velocities ; P, Q, R, normal components of pull (per unit 

 area) on interfaces respectively perpendicular to a, y, z ; and 

 S, T, U respectively the tangential components of pull as 

 follows : — 



of= pull parallel to y on face perpendicular to z, -? 



{: 



» 



» 



z 



)) 



» 



X 



)1 



» 



X 



V 



» 



y 



9. For an isotropic solid, us 

 above (§ 3), we havef 



' y ■ (5). 



ng Green's notation A, B as 



* Thomson and Tait, 'Natural Philosophy,' § 697. 



t Thomson and Tait, < Natural Philosophy,' § '693 and § 670 (6). 



