Doctrine of Extraneous Force. 121 



and 



a = </ (fty) [ar(mv + n/ju) + cr 2 /MV~] ; b = V (ya) [a {rik + Iv) + <t 2 v\~] ; 

 c =^( a /3)[<r(//i + m\) + <rV] . . . (12), 



gives three real positive values for p, the square roots of which 

 are the required principal strain-ratios. 



9. Entering now on the dynamics of our subject, remark 

 that the isotropy (§1), implies that the work required of the 

 extraneous pressure, to change the solid from its unstrained 

 condition (1, 1, 1) to the strain (a, ft, 7), is independent of 

 the direction of the normal axes of the strain, and depends 

 solely on the magnitudes of a, ft,y. Hence if E denotes its 

 magnitude per unit of volume ; or the potential energy of 

 unit volume in the condition (a, ft, 7) reckoned from zero in 

 the condition (1, 1, 1) ; we have 



E=t(«,/3, 7 ) (13), 



where ty denotes a function of which the magnitude is un- 

 altered when the values of a, ft, y are interchanged. Consider 

 a portion of the solid, which, in the unstrained condition, is a 

 cube of unit side, and which in the strained condition [a, ft, 7), 

 is a rectangular parallelepiped V a. . V ft . Vy. In virtue of 

 isotropy and symmetry, we see that the pull or pressure on 

 each of the six faces of this figure, required to keep the 

 substance in the condition (a, ft, 7) is normal to the face. 

 Let the amounts of these forces per unit area, on the three 

 pairs of faces respectively, be A, B, 0, each reckoned as 

 positive or negative according as the force is positive pull, or 

 positive pressure. We shall take 



A+B+C=0 (14), 



because normal pull or pressure uniform in all directions pro- 

 duces no effect, the solid being incompressible. The work done 

 on any infinitesimal change from the configuration (a, ft, 7) is 



A V{fty)d{ V «) + B V (y*)d( */ ft) + C V(*/3M ^7)> ~ 

 or (because ufty=l) 



-da+^dB+^dy 



K15). 



10. Let 8a, 8ft, 8y be any variations of a, ft, 7 consistent 

 with (2), so that we have 



(a + 8a)(ft + 8ft)(y + 8y) = l-) 

 and I . . (16). 



afty=l J 



Now suppose Sen, 8ft, 8y to be so small that we may neglect 



