165] 



STRAIN- QUADRICS. 



433 



that is, the axes of the ellipsoids i(j, ij> are proportional to the squares 

 of the axes of the quadrics tp and g>'. By means of either pair of 

 quadrics <p, ^ or g/, tf we get a complete representation of the linear 

 vector function, the directional relations heing given by 9? or 9' as 

 above described, the quantitative relations by ty or tf (Fig. 144). 



Either quadric has three principal axes given by the three real 

 roots of the determinantal cubic. These are called the principal axes 

 of the strain. Conjugate diameters remain conjugate, since parallel 

 lines remain parallel and bisected by corresponding lines. Accordingly 

 the principal axes of the strain -ellipsoid must have been before the 

 strain conjugate diameters of a sphere, and therefore mutually 

 perpendicular. As the axes of the ellipsoid are the only set of 

 mutually perpendicular conjugate diameters, there is no other set of 

 mutually perpendicular lines which remain so after the strain. 



Fig. 144. 



The equation 5) to determine the direction of lines which 

 maintain their direction after the strain becomes for pure strain, 

 represented by equations 9), the determinantal equation for the axes 

 of the quadric qp. We have shown in Note IV that this equation 

 always has three real roots and 'that the corresponding directions are 

 mutually perpendicular. Thus the axes of a pure strain have their 

 directions unaltered by the strain. A pure strain is therefore called 

 irrotational. (It is to be noted that the axes are the only lines which 

 are not rotated.) 



If the quadrics are referred to their principal axes the equations 

 assume the following simple forms from which the geometrical rela- 

 tions are easily seen, 



WEBSTEB, Dynamics. 28 



