430 



IX. DYNAMICS OF DEFORMABLE BODIES. 



Accordingly the left-hand member of 7) must vanish. If r s is 

 perpendicular to t\r 2 its coordinates must be proportional to their 

 vector -product. Thus we may write 7) as 



Inasmuch as the order of suffixes 1, 2, 3 is indifferent, if the 

 three vectors r 19 r 2 , r s are to be mutually perpendicular, equation 8) 

 must be satisfied by the components of all three. This can be true 

 only if we have 



that is, the determinants of the substitutions 1) and 2) are symmetrical. 

 In this case the linear vector -function is said to be self - conjugate, 

 and a strain represented by such a function is called a pure strain. 



165. Self -conjugate Functions. Pure Strain. We will 

 consider this important case in detail. Adopting a symmetrical 



notation, let us write . 7 



= ax + hy + gs, 



9) y' = hx + l)y -f fz 9 

 z 1 = gx 4- fy+ cz. 



If by cp we denote the homogeneous quadratic function 



10) fp = ax z -\- by* -f # 2 + %fy& + 2g2x + 2hxy, 

 equations 9) may be written 



so that the vector OP' (Fig. 143) is parallel 

 to the normal at the point P, whose co- 

 ordinates are x,y,z, lying on the quadric 

 cp = + E 2 } where E is a constant introduced 

 merely for the sake of homogeneity. In 

 like manner calling 



Fig. 143. 



11) (p' = 

 -f 

 equations 2) are 



-f 2 



X 



12) 



so that OP is parallel to the normal at P', whose coordinates are 

 x', y', z', a point on the quadric <p' = + B 2 . 



