Theory of Elasticity. 275 



placement relations in this case, for there are no displacements 

 corresponding to the primary strains. Thus, since the equa- 

 tions of equilibrium alone are insufficient, primary stresses and 

 strains are not mathematically determinable, a fact quite in 

 accordance with our knowledge of their possibility of infinite 

 variation. They are merely limited, not defined by the equa- 

 tions of equilibrium. They must satisfy these, they need 

 satisfy no others. Any of the infinitude of solutions of these 

 equations offer equally possible values (physical capacity for 

 resistance apart). 



Although they are not mathematically definable, primary 

 stresses and strains may be physically defined. Primary 

 stresses are those which, applied to a portion of a body when 

 removed from the body, and at the same temperature, would 

 reduce that portion of the body to the form and proportions it 

 had when in the body, when the body was free from all bodily 

 and external forces, — while primary strains are the correspond- 

 ing strains. By cutting a body into more or less numerous 

 pieces and measuring the changes in these pieces resulting from 

 their release from the body, primary stresses and strains could 

 be experimentally determined. But except where they have 

 been systematically introduced or created, or can be deter- 

 mined by acoustic or optical methods, without such cutting 

 up they must remain, in general, essentially undeterminable. 



Since primary stresses and strains are not mathematically 

 definable and actual stresses and strains have primary stresses 

 and strains as one component,* actual stresses and strains are 

 essentially mathematically indeterminate. The theory of elas- 

 ticity alone does not, can not define the actual state of stress 

 and strain in any real body. 



where (e)(f) (g) (a) (b) (c) are the primary strains. Substituting from (6") in (1") 

 and (2") we obtain 



For interior points. For surface points. 



X d p + f ,( 2 d p + d ^ + d p)=0 1 etc (7") M(A) + /z[2Z(e) + ro(c) + n(&)] = 0,etc. (8") 

 ox \ ox oy oz ' 



as the equations limiting the primary strains. 



* Absolute stresses are the sum of primary stresses and the changes in stress 



due to applied forces, and absolute strains are the sum of the corresponding 



primary strains aud changes in strain or 



~P=(P) + P ~S~=(S) + S,etc.(ll) ~T=(e) + e cT=(a) + a, etc. (12) 

 whence simply by addition of (7') and (7") and of (8') and (8") we get 

 For interior points. For surface points. 



l d -£ + f t{2^-+ ^- + ~) = - P X,etc.m MA+f*(2U +m~c~ + nb),=F, etc. (8) 

 dx \ dx dy dz / 



for the equations limiting the actual or total strains in elastic isotropic bodies 

 subject to Hooke's law. So further limits may, in general, be set to them. 



