372 J. W. Gibbs — Equilibrium of Heterogeneous Substances. 



H — 



(442) 



It will be observed that to change the signs of two of the quantities 



/•j, r^, /"g is simply to give a certain rotation to the body without 



changing its state of strain. 



Whichever supposition we make with respect to the axes TT',, U^, 



U,^^ it is evident that the state of strain is completely determined by 



the values E^ F^ and H^ not only when we limit ourselves to the 



consideration of such strains as are consistent with the idea of solidity, 



dot dz 



but also when we regard any values of -y-, ,...—, as possible. 



Approximati'oe Forniidce. — For many purposes the value of fy, for 



an isotropic solid may be repi'esented with sufficient accuracy by the 



formula 



fv, =: i' + e' E-\-f' F-\- h' II, (443) 



where ?'', e',/", and K denote functions of ;;v, ; or the value of //v/ by 

 the formula 



//•v, = / + e E-^fF-^r h H, (444) 



where ^, e,/*, and h denote functions of t. Let us' first consider the 

 second of these foi-mulae. Since E, F, and H are symmetrical func- 

 tions of r,, rg, ^3, if ij\-, is any function of ^, E, F, II, we must have 



dij.'v, dil'y, di/'v, 



dr. 



dll'y, 



dr^ 



dr 



d^>[\-, 

 dr,^ ' 



d^tv, 



(445) 



d^'y,_d^^y, 



dr^^ dr„^ 

 d^if\, _ d^ij-y, 

 dr^dr.j dr^dr^ 



whenever r^:=r2=r^. Now ^, e,,/', and h maybe determined (as 

 functions of t) so as to give to 



dr^dr ^ 



\ 



J 



'/v. , 



di\\., d-if^, d- ij\, 

 dr^ ' dr^^ ^ dr^dr^ 



their proper values at every temperature for some isotropic state of 

 strain, which may be determined by any desired condition. We 

 shall suppose that they are determined so as to give the proper 



