702 Dr. C. V. Burton : Notes on 



But in this case, by (3) <l>, M*, . . . all vanish ; hence 



0=B 1= B 2 = , 



and (5) becomes 



"W = (a homogeneous quadratic function of </>, yfr, . . .) 



+ iA0 2 ; (7) 



so that, to our degree of approximation, the increment of 

 potential energy due to a change of c/>, i|r, . . . . from one set 

 of values to another, is independent of 0. We are thus led to 

 deny the possibility, by means of a locked distribution of 

 infinitesimal strain, of conferring on the medium any new 

 lability, in virtue of which no increase of potential energy 

 would be involved in the superposition of certain other types 

 of strain compatible with the locked condition. 



12. The proof here given that no change of elastic proper- 

 ties is caused by a locked distribution of infinitesimal strain 

 is of very general application. Nothing need be assumed as 

 to the nature of the coordinate 0, except that it involves some 

 violation of the constraints which normally are imposed on 

 the medium, and in virtue of which the values of the co- 

 ordinates 4>, i/r, . . . . suffice to render the whole configuration 

 determinate. When the discussion is not limited to small 

 strains, so that stress-strain relations need no longer be linear, 

 the result obtained in § 11 ceases to be applicable without 

 modification. But this is evidently far from implying that 

 mobility is assured to a locked strain-figure, provided merely 

 that the strains are great enough to exceed the limits of 

 linearity. Rather is it evident, from consideration of the 

 simple case where linear relations hold good, that in a 

 thoroughly stable medium locked strain-figures are not rightly 

 constituted for mobility. 



13. The general difficulty of conceiving a freely mobile locked 

 strain-figure is perhaps more forcibly realized when we begin 

 to trace the kinematical consequences of mobility. In rela- 

 tion to the distribution of displacement constituting the 

 strain-figure, the surface which we regard as having been the 

 seat of a cutting and reuniting operation is a surface of dis- 

 continuity ; the displacement in general suffering a finite 

 change of value as we pass from a point on one side of the 

 surface to an infinitely near point on the other side. If, then, 

 we suppose the locked strain-figure to be moving through the 

 medium, and consider the motion of any point of the medium 

 lying in the path of the surface of discontinuity, we realize 

 that, just at the instant when the discontinuity passes, the 

 point in question suffers impulsively a finite change of 

 position. Thus the assumed motion of the locked strain-figure 



