﻿Elastic Equilibrium under Surface Tractions, 507 



These three equations take the form 





d# 2 o^ 1 



= — *« -^"2 "1 -.2 



-- 1 ' 2 a* 2 -" 1 — aF~ ' 



so that . . . (15) 



with two similar equations, and 



+ "3(^-^2)} = 0. (16) 



Hence Vi(0 2 — #3) + ^2(^2 — #i) + ^2(^1— #2) is an Ellipsoidal 

 Harmonic, from which Q x — 2 , 6 2 — 6- 6 can De deduced. 



In a previous paper, I have dealt with the simpler case 

 for an isotropic body, when v 1 = v 2 = v 3 = n and 6\ — 6 2 , 2 —d 3 

 are Spherical Harmonics. 



The present equations apply to crystals having three 

 orthogonal planes of elastic symmetry. 



The investigation seems interesting because it passes some- 

 what outside the range of elastic equilibrium, even if it is 

 ultimately confined within that range. If we regard a piece 

 of sound material intended to serve as a test-piece we cannot 

 consider its potential energy of strain to be precisely in the 

 form given by (10), although by judicious working it tends 

 to approach that form. The actual potential energy has pro- 

 bably a form such as that in (3), until it has been worked. 

 The effect of "working" simplifies the form of V in a 

 manner perhaps comparable with the algebraic discarding 

 of constants inconsistent with elastic equilibrium. If this 

 comparison is not unreasonable, we may venture to extend 

 the idea, and to imagine that an excessive exertion of trac- 

 tion may again alter the form of the potential energy, and 

 introduce into V terms inconsistent with elastic equilibrium 

 and may, if continued, lead to rupture. At any rate, the 

 theory of rupture must lie outside the range of elastic equi- 

 librium, though not necessarily outside the elastic stress- 

 strain relations. 



