Theory of Elasticity. 



289 



may occur, at any rate in certain cases of elastic instability, 

 the possible states of equilibrium may be several. 



Among the limiting conditions of changes in stresses and 

 strains, — the ordinary stresses and strains of the theory of elas- 

 ticity, — are what are known variously as the equations of com- 

 patibility or of continuity.* These simply state relations 

 which must subsist between the strains in order that the sum 

 of the products of the strains in a given direction with the dis- 

 tances, along any path whatsoever from one point to a second 

 point, shall be a quantity independent of the path taken, — the 

 displacement in the given direction of the one point with 

 respect to the other. These equations have no reference to 

 and do not apply to either primary or absolute strains, as will 

 be found by attempting to apply them to the primary strains 

 of the illustrations. The displacement of one point with 

 respect to another accompanying the generation of primary 

 strains involves flows or slips, and strain sums alone will not 

 express it nor will they be equal along different paths between 



ness is also removed, since uvw are given at the surface, and if three points of a 

 rigid body be moved in a given manner the displacement of all points is determi- 

 nate." 



Examining the above demonstration, we note that it assumes that the stress 

 equations of equilibrium have solutions in terms of displacements, which can 

 only be true provided the stresses are fuoctions of the displacements. So 

 Kirchhoff's demonstration only applies to stresses that are functions of the dis- 

 placements, that is to say to changes in stress due to applied forces. This is 

 further confirmed by the use he makes of Green's transformation. If we inte- 

 grate by parts we should simply get 



()■:>: 



Mi-C- 



dU' dT 

 dy dz 



I dx dy dz 



■ff 



\u\lF 



mU'+nT) 



ds 



-M\ 



du nl dv' _. 

 P' + — Q + 



Mm 



. c' are the derivatives 



+ : — I of the dis- 



dx dy dz 

 dx ()y 



■t term only reduces to the form given by Kirchhoff providing that the 

 du' dv' /dvj dv' 



dx dy- \ dy dx 



placements and the stresses P' are the rates of change of the energy 



function W with respect to these strains. This is not true of primary stresses 

 and strains but only of changes in stress and strain due to applied forces. 

 *Love gives them in his Theory of Elasticity, Art. 66 (6), vol. 1 as follows: 



and this las 

 strains e'f . 



£)* 



>rovidit 



(dvj dv' \ 

 dy dx J 



d*e 



*v 



dz' 1 

 d*g 

 ~lx* 

 and these relations 

 must conform to. 



dx 2 ' 

 dy* ~ 



the 



d*c 

 dxdy 

 d*a 



d 2 e 

 dydz 

 d*f 

 dzdx 

 d*g 



dydz 

 d' 2 b 

 ' dzdx 

 changes in strain 



2 T-T + 

 dxdy 



d*a 



~dxf 

 d*b 



~W 



d*c 



"dV 



d*b 

 ' dxdy 



d*c 

 dydz 



d*a 



dzdx 



d^c -, 

 dzdx I 

 6*a 



dxdy { 



dydz J 



(42) 

 (43) 



accompanying any elastic distorsion 



