210 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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 squares of the ratios of linear elongation and of 

 superficial enlargement, for all possible directions in the unstrained 

 solid. 



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

 determined by E, F, and H, instead of E, 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 systems 

 of co-ordinate axes X, F, Z, and X', F', Z' y are either identical or 

 such as are capable of superposition, which it will always be con- 

 venient to suppose, the determinant H will always have a positive 

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

 possible to give to x, y, z such values as functions of x', y', z that H 

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



x=x', y = y', z=z'. (440) 



This will give H= 1, while 



x=x', y = y', z=*z' (441) 



will give #=1. Both (440) and (441) give # = 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 

 a method 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, F, and H to repre- 

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

 in the state (x', 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 positive 

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

 be the nature of the parallelepiped in the state (x, y, z) which corre- 

 sponds to the cube dx', dy', dz' and is determined by the quantities 



-r->, ... -j- f , it may always be brought by continuous changes to the 



d/x dz 



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

 be parallel to the axes of X 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 parallelepiped 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 



