288 Cilley — Fundamental Propositions in the 



examination shows that this is not the case, for we lind the 

 demonstration has reference only to changes in stress and strain 

 due to applied forces, that it depends on suppositions applica- 

 ble only to these changes, and therefore that it has no bearing 

 on and in no wise limits actual or total stresses and strains. A 

 misunderstanding among elasticians as to the scope of this 

 theorem of Kirchhoff is probably one reason, for their neglect 

 to develop the general theory of primary stress and strain. 

 We may note in this connection that the validity of Kirchhofi's 

 demonstration (as here given) is limited to cases of relatively 

 small displacements and to bodies subject to Hooke's law. It 

 is not evident that the latter limitation is essential, for appar- 

 ently the distortions of bodies not subject to Hooke's law, due 

 to applied forces, are quite as singly determinate as those of 

 bodies which are subject to that law. The former limitation is 

 apparently quite necessary, for, where great displacements 



dP dir <)T_ _ 



dx dy ()z 



and three such boundary conditions as 



IF + mlT + nT =0 (47) 



where P . . . . are the stresses corresponding to these displacements. 

 Now, by Green's transformation 



JJf\ "'OS + w + w) + + \ dxdydz 



= ff\ u'(lP +mW +nT') + + .....' dS 



/'/'r/,«W' .. tfW , ,iW '\, j , 



-J J J ( e — e > +f ^r + +e -&-Y* 



where e'f . . . . c are the strains corresponding to the displacements u'v'w'. 

 Hence 



JJJ [ e M +f ~6f + +C ^y^ydz = 



m 



But from the form of W as a positive quadratic function, we know that the 

 expression under the integral sign is 2 W so that the integral is a sum of posi- 

 tive terms which can vanish only when e ==/ =....= c' = 0. Thus the dis- 

 placements {u'v'w') are such as are possible for a rigid body, and the solution is 

 only indeterminate to the extent of such displacements. 



2°. Supposing the bodily forces and surface displacements given, we take as 

 before, two solutions UyVxiu^ u 2 v<iW 2 and form their differences u'v'w', then u'v'w 

 satisfy stress equations like 



dP' dU' ()T 



^— + "I- + -T- = ° 

 ox oy dz 



and boundary conditions u = v' = w' = at the surface. Thus we find that 



rrri > d w *, 6 w , 6W/ \ 



JJJv« +i v- + +c -^) 



dx dy dz = 



and hence e' = f= . . . c'= and the displacements are only indetermi- 

 nate to the extent of displacements possible for a rigid body. This indeterminate- 



