STRAINED ELASTIC SOLIDS 497 



regard the three quantities dn/dan, Qr^/dan, drs/dan as deter- 

 mined by the resulting three simultaneous equations in these 

 quantities (determined, i.e., in terms of the Upq coefficients). 

 Similar statements are true for any of the partial differential 

 coefficients dri/da„y, drt/dapq, dr^/dapq. These are of course 

 correct values and have nothing to do with the approximation 

 to \pv made in [444]. Now Xx' is determined as we know by 

 the equation 



), near the top of page 



204.) Since d(ri — ro)"/dan = n(ri - roy~^dri/dan, etc., we can 

 express Xx' as an ascending series in the quantities ri — ro, 

 T2 — To, rs — To, and since the true and the approximative series 

 for xpv' agree to the second degree, the true and approximative 

 series for Xx' will agree to the first degree, and the error in 

 Xx> involved in using the approximative series will be of the 

 order of magnitude of the squares of n — ro, ^2 — ro, r^ — ro. 

 On pages 213, 214, e, f, h are determined in terms of the bulk- 

 modulus and the modulus of rigidity. These two moduli, as we 

 have mentioned earlier, possess physical significance only in so 

 far as Hooke's law is obeyed; and this, as experiment demon- 

 strates, restricts the range of stress allowable from the unstressed 

 state at a given temperature. Gibbs' calculations on page 

 213 are limited by this consideration, as he himself expressly 

 admits; for he indicates that his moduli are determined for 

 "states of vanishing stress," and in the final results he goes to 

 the limit at which n = r2 = rs = ro; ro as before being the 

 uniform ratio of elongation due to the change from the tem- 

 perature for the state of reference (regarded as unstressed) 

 to the temperature indicated by t. The formula for the bulk- 

 modulus in [448] we have discussed earlier. To use it we must 

 express p as a function of v and t. Consider a mass of the solid 

 which has unit volume in the state of reference. It is subjected 

 to the change of temperature which gives it the volume ro^ 

 It is now subject to uniform pressure p which gives it a uniform 



