STRAINED ELASTIC SOLIDS 443 



But a glance at (37), or [355] of Gibbs, shows us that, if no 

 heat is imparted and only an varies, the increase in energy of 

 the element is 



Hence as work done is equal to energy increase the force just 

 referred to is 4:r]'^'Xx', or Xx> per unit of area in the state of 

 reference. The symbolism clearly indicates the physical 

 signification; the accented x' in the suffix indicates that the force 

 is estimated on an area which was perpendicular to OX' in the 

 unstrained state and was equal to the unit of area in that state. 

 The unaccented X, to which x- is the suffix attached, indicates 

 that the force is a component in the direction OX. The force 

 of course only exists in the strained state, since the reference 

 state is assumed as an unstrained state, that is, one in which the 

 stress-constituents vanish. (See the remarks on this on page 418.) 

 It is clear from this (quite apart from the type of equations 

 connecting Xx', ... Zz' with Xx, . . . Zz which are indicated 

 above) that Xx is quite distinct from Xx'', for Xx is the force 

 across a face which is perpendicular to OX in the state of strain 

 estimated on an area which is equal to the unit area in that 

 state; it is however, like Xx', a component in the direction OX. 

 Similar differences can be drawn between the other com- 

 ponents of stress in the two systems of coordinates. From this 

 it can be perceived that because Yz = Zy it is not of necessity 

 true that Yz' = Zy. It should be observed that these results 

 do not depend on the fact that one may choose the axes OX, OY, 

 OZ not to coincide with OX', OY', OZ'; for even if they were 

 made to coincide the symbol Xx, for example, could not be 

 made to do double service, on the one hand for a component 

 parallel to OX of a force across an area which was unit area in 

 size and was perpendicular to OX, and on the other hand across 

 an area which is unit area in size and is perpendicular to OX. 

 Thus the double naming of the axes is of service even when they 

 are regarded as coincident. This is a justification for Gibbs' 

 apparently pointless complication of procedure. Only if the 

 state of strain is regarded as being little removed from the state 

 of reference can we assume that an approximate equality may 



