498 RICE ART. K 



elongation with the ratio Vi in all directions as compared with the 

 state of reference at the original temperature, so that its volume is 

 now ri^ (n = rz = ra). Thus E = Sri^ = Sv' ; P = 3ri* = Sv^; 

 H = r^ = V, and so we arrive at [451]. By equation [88] from 

 the earlier part of Gibbs' discussion we obtain the general expres- 

 sion for p in [452] in any state of uniform stress small enough to be 

 consistent with Hooke's law. Differentiation gives us [453], 

 and an approach to the limit at which v = r^ gives us the result 

 [454]. 



The writer is unable to justify the equation [449] as it stands; 

 as far as he can judge it ought to read 



dXy' 



R = ro 



da 



12 



To see this, let us consider the matter from the point of view of 

 the ordinary treatment of isotropic solids in the text-books of 

 elasticity. Limiting ourselves to strains so small that Hooke's 

 law applies, the modulus of rigidity is defined as the common 

 value of the quotients 



Yz ^x Xy. 



fi U U 



The quantities fi, /e, /e are the shears of the lines parallel to 

 axes of reference (the same axes for the state of strain as for the 

 state of reference). As we saw in our discussion the value of 

 /a, for example, is ee/(ele2)^ although it can be replaced by an 

 approximation Cn, + 621 for very small strains. This, of course, 

 implies that changes of temperature are not involved. Let us, 

 however, consider the situation which arises when the state of 

 strain is at a temperature t, different from the temperature of 

 the state of reference. The definition of the modulus of rigidity 

 at temperature t must of course involve the shears of the axes 

 from an unstressed state also at that temperature, that is, a 

 state in which all lengths are elongated in the ratio ro as com- 

 pared with the state of reference. The definition of R is still 

 Xy/fi (say), and /e is still 66/(6162)^ But we have to be careful 

 about the approximation. Let us recall the definitions of 



