370 J. W. Gibbs — Equilibrimn of Heterogeneous Substances. 



single-valued functioji of ?/v, , E, E, and M. The determination of 

 the fundamental equation for iaotropic bodies is therefore reduced to 

 the determination of this function, or (as appears from similar con- 

 siderations) the determination of il\-, as a function of t, JE, F, and H. 



It appears from equations (4;:i9) that E represents the sum of the 

 squares of the ratios of elongation for the principal axes of strain, 

 that ^rejjresents the sum of the squares of the ratios of enlargement 

 for the three surfaces determined by these axes, and that G repre- 

 sents the square of the ratio of enlargement of volume. Again, equa- 

 tion (432) shows that E represents the sum of the squares of the 

 ratios of elongation for lines parallel to -ST', Y', and Z'; equation 

 (434) shows that ^represents the sum of the squares of the ratios of 

 enlargement for surfaces parallel to the planes X'- Y\ Y'-Z', Z'-X' ; 

 and equation (438), like (439), shows that G represents the square 

 of the ratio of enlargement of volume. Since the position of the 

 co-ordinate axes is arbitrary, it follows that the sum of the squares of 

 the ratios of elongation or enlargement of three lines or surfaces 

 which in the unstrained state are at right angles to one another, is 

 otherwise independent of the direction of the lines or surfaces. 

 Hence, ^E and ^F are the mean sqiiares of the ratios of linear elon- 

 gation and of superficial enlargement, for all possible directions in 

 the unstrained solid. 



There is not only a practical advantage in regarding the strain as 

 detei-mined by E, F, and H, instead of -£', F, and G, because H is 



more simply expressed in terms of -^, ,...-=-,, but there is also a 



certain theoretical advantage on the side of E, F^ H. If the sys- 

 tems of co-ordinate axes X., I^, Z, and A"', I"', Z\ are either iden- 

 tical or such as are capable of superposition, which it will always be 

 convenient to suppose, the determinant H will always have a posi- 

 tive value for any strain of which a body can be capable. But it is 

 possible to give to a*, y, z such values as functions of x' ., y\ z' that H 

 shall have a negative value. For example, we may make 



a; = a;', y =: y', z=i -^ z'. (440) 



This will give 11:= — 1, while 



x-=ix\ y^=y\ z=^z (441) 



will give II=z 1. Both (440) and (441) give 6^ = 1. Now although 

 such a change in the position of the particles of a body as is repre- 

 sented by (440) cannot take place while the body remains solid, yet 



