700 Dr. C. Y. Burton : Notes on 



maintained by the application of an ideal forcive may be 

 called a forced strain ; a strain-figure specified as in the pre- 

 ceding paragraph, and requiring therefore no application of 

 force to maintain it, being called free. It is proposed to 

 restrict the designation " free " to a strain-distribution so 

 constituted that it could be (ideally) brought into existence 

 by a continuous process, involving no cutting or slitting of 

 the medium, or any deviation from perfect elasticity (or 

 quasi-elasticity). On the other hand, distributions of strain 

 have been conceived which can only be brought into being 

 by some such discontinuous operation as slitting the medium, 

 and welding together or reuniting the slit surfaces after im- 

 posing on them a relative twist, or otherwise changing their 

 presentation to one another. Such distributions, though self- 

 sustaining, are not in our sense free, and in conformity with 

 general usage they may be termed locked. 



11. Locked Strain-figures are not in general mobile. — We 

 have seen in § 9 that if a strain-figure is capable of existing 

 in the " free " state, there will at least be no static resistance 

 to displacing it with respect to the medium. In discussing 

 the possible mobility of locked strain-figures, we have to 

 remember that a determinate displacement of any strain- 

 figure with respect to the medium is equivalent to the super- 

 position of an additional strain-distribution. Free mobility 

 thus implies that there are certain special strains which can 

 be superimposed on the strain-figure without the expenditure 

 of work ; that is to say, in regard to such special strains the 

 equilibrium of the medium is labile. Now it might be ex- 

 pected from general considerations that, to a first order, the 

 stability of a medium is not affected by locally cutting it 

 asunder and reuniting the cut surfaces after altering their 

 presentation to one another: a conclusion which can in fact 

 be verified by a simple analysis. 



Before any cutting has taken place, let the state of strain 

 of the medium be defined by the values of the coordinates 

 <j>, yjr, . . . ,, all supposed independent. Now let the medium, 

 originally unstrained, be cut asunder and the cut surfaces 

 so reunited as to involve a condition of strain ; the specifica- 

 tion of the cut and of the displacements introduced being 

 determinate, so that the strain remaining when the medium 

 is subsequently left to itself is likewise determinate, and is 

 definable by assigning a definite value to a suitably chosen 

 coordinate 0, with simultaneous zero values for <j>, yjr, . . . . 

 There are precisely the same kinematic possibilities for the 

 uncut medium, in which 6 is constrained to be zero, and 



