126 Me. R. F. Gwyther on 



(j3) There is no linear vector function of the second 

 differential coefficients of u, v, and w, which satisfies the 

 conditions except that whose components are 



d fdu dv dw\ /d 2 u d 2 u d 2 u\ 

 dx\dx dy dz) \dx 2 dy 2 dz 2 J' 



From this latter we see that the equations of equilibrium 

 of an elastic solid, on the hypothesis of isotropy, have their 

 form fixed by these considerations, and that, as I have 

 indicated earlier, these are exactly the forms which are 

 derived from a vector by the use of Hamilton's operator. 



3, I shall proceed to draw some other examples from the 

 subject of elasticity, and, finally, to extend the conditions 

 beyond those for isotropy. 



If we confine our attention to the first differential 

 coefficients of a vector function, which I shall look upon as 

 the displacement at a point in an isotropic solid, and if we 

 replace these differential coefficients by the elements of the 

 strain (e, f, g, a, b, c) and the components £, rj, Z, of the 

 rotation, the forms of the operators are sufficiently 

 indicated by 



d d\ „, _ v d d d 



dc 



( d d\ n . „. d d 7 i 



yd d 



+ v^ ■- - (7) - 



From these we obtain easily the usual invariants of the strain 



e+f+9, etc. 

 and, if we take the origin at the point and consider also a 

 point whose vector components or co-ordinates are x, y, z, we 

 find the usual strain quadric as a covariant 



Covariants of a more general character will be noticed 

 later. The potential energy of the strain (V) reckoned per 

 unit volume, must be an invariant, and must, therefore, be a 

 function of the invariants of the strain. Following Green, 



