Theory of Elasticity. 287 



The fact that changes of stress and strain and therefore dis- 

 tortions due to applied forces are practically independent of 

 pre-existing stresses and strains has previously been mentioned 

 but not explained. The reason for this fact is very simple, 

 being the linear character of the stress-strain relations, the con- 

 sequences of Hooke's law, — " stress is proportional to strain." 

 Where Hooke's law closely holds, that is to say, within the 

 elastic limits of most bodies to which it is attempted to apply 

 the mathematical theory of elasticity, it makes no sensible dif- 

 ference in the elastic distortions due to applied forces whether 

 primary stresses and strains exist or not. This probably 

 explains their entire neglect by the theoretical elastician, for 

 he has limited himself chiefly to investigations precisely within 

 these limits of "proportional stress and strain, with the deter- 

 mination of distortions consequent on the application of given 

 forces, or the converse, as his one aim. Only when primary 

 strains were very great, as possibly in some castings, would 

 their presence noticeably affect distortions due to applied 

 forces, for the unstrained lengths to which all strains are 

 referred would then appreciably differ from the strained 

 lengths. But with metals within the elastic limits this never 

 becomes a very important factor. The real importance of the 

 existence of primary strains is that they cause a body to 

 become overstrained at a different time and place from what 

 would ordinarily be expected. 



The idea of innumerable possible solutions of the general 

 stress or strain equations of equilibrium at first sight may seem 

 contrary to Kirch hofPs celebrated demonstration of the unique- 

 ness of solution of the equations of elasticity.* But a closer 



*Kirchhoff's demonstration of the uniqueness of solution of the equations of 

 elasticity as given in Love's Theory of Elasticity, vol. i. Art. 66(e), pp. 123-4 is 

 "(e). If either the surface displacements or the surface tractions be given the 

 solution of the general equations of equilibrium is unique. 



1°. Supposing the bodily forces and the surface tractions given, then, taking 

 Wa quadratic function of the six strains, we have 



be 

 also the general equations of equilibrium are three such as 



6P 6U oT _ n ,.., 



-t— + -r- + -T- + f>X= (44) 



ox oy oz 



and the boundary conditions are three such as 



lP + mU+nT = F (45) 



If possible suppose there are two different solutions of these sets of equations, 

 and that the corresponding displacements are UiVitOi and u 2 v 2 w 2 in the two solu- 

 tions. Then, writing 



u'=Ui—U 2 v'=Vi—V 2 w'=tv 1 — w- 2 



we see that u'v'w' are a set of displacements which satisfy three such differential 

 equations as 



