STRAINED ELASTIC SOLIDS 



463 



9 Ax' dXr' dXi 



JO 



'dYx' 9Fy' dY 



+ 



(dZx' dZy' dZz' A I . . . 



+ 



(a'Xx' + ^'Xy' + t'Xz') + av 



D£ 

 Ds' 



8x 



+ 

 + 



Dsl 



+ 



(a'7.v' + /3'Fk. + 7'FzO + pp j^A 8y 

 (a'Zx' + ^'Zy> + t'-^z') + TP;^J 5z\ds' 

 ev - tr]v' + pvv - 2 (mi^i) ^^' ^^' = ^■ 



This is equation [369] written in full. 



Since, in the volume integrals, 8x, by, 8z are arbitrary varia- 

 tions, the expressions multiplying them must be zero at all 

 points of the solid in order that [369] may be true for any rela- 

 tive values of 8x, 8y, 8z. Thus we arrive at equations [374]. 

 In the second integral of our rewritten [369] the expressions 

 multiplying 8x, 8y, 8z respectively must also be zero at all 

 points of the surface for the same reason. Thus we arrive at 

 equations [381]. There remains only the third integral in the 

 rewritten [369]. If 8N' is quite arbitrary, i.e., if crystal- 

 lization on the solid and solution from it are both possible we 

 must accept the truth of [383] ; but if the values of 8N' can only 

 be chosen arbitrarily from infinitesimal negative numbers, i.e., 

 if solution only is possible, we justify only the wider conclusion 

 [384]. 



At the bottom of page 190, Gibbs makes a passing reference 

 to the stress-constituents Ax, Xy, . . . Zz i.e., the constituents 

 measured across faces perpendicular to the same axes as those 

 which indicate the directions of the thrusts or pulls involved in 

 the definitions of the constituents. His proof of the equality 



