J. TK Gihhs — Equillhrimn of Heterogeneous Substances. 871 



a metliod of representing strains may be considered incomplete, 

 which confuses the cases represented by (440) and (441). 



We may avoid all such confusion by using E, I^] and II to repre- 

 sent a strain. Let us consider an element of the body strained which in 

 the state (ar', y', z') is a cube with its edges parallel to the axes of 

 X', Y', Z\ and call the edges dx\ dy' , dz according to the axes to 

 which they are parallel, and consider the ends of the edges as posi- 

 tive for which the values of x, y\ or z' are the greater. Whatever 

 may be the nature of the parallelopiped in the state (^■, y, z) which 

 corresponds to the cube ffe', dy\ dz' and is determined by the quanti- 



ties -=—, , . . . -^ , it may always be brought by continuous changes 



to the form of a cube and to a position in which the edges dx\ dy' 

 shall be parallel to the axes of JC and Y, the positive ends of the 

 edges toward the positive directions of the axes, and this may be done 

 without giving the volume of the parallelopiped the value zero, 

 and therefore without changing the sign of H. Now two cases are 

 possible ; — the positive end of the edge dz' may be turned toward the 

 positive or toward the negative direction of the axis of Z. In the 

 first case, H is evidently positive ; in the second, negative. The 

 determinant H y^Wl therefore be positive or negative, — we may say, 

 if we choose, that the volume will be positive or negative, — according 

 as the element can or cannot be brought from the state (a-, y, z) to the 

 state («', y', z') by continuous changes without giving its volume the 

 value zero. 



If we now recur to the consideration of the principal axes of strain 

 and the principal ratios of elongation rj,r2, ^3, and denote by f/^, 

 f/2, C^3 and TT^', f/g', U^' the principal axes of strain in the strained 

 and unstrained element respectively, it is evident that the sign of r^, 

 for example, depends upon the direction in U^ which we regard as 

 corresponding to a given direction in U^'. If we choose to associate 

 directions in these axes so that r,, rg, r^ shall all be positive, the 

 positive or negative value of i? will determine whether the system of 

 axes ?7,, U^, U^ is or is not capable of superposition upon the sys- 

 tem C/"/, U2, U^' so that corresponding directions in the axes shall 

 coincide. Or, if we prefer to associate directions in the two systems 

 of axes, so that they shall be capable of superposition, cori-esponding 

 directions coinciding, the positive or negative value of II will deter- 

 mine whether an even or an odd number of the quantities /-j, r,, r, 

 ai-e negative. In this case we maj^ write 



