428 



IX. DYNAMICS OF DEFORMABLE BODIES. 



No constant terms are included because a displacement represented 

 by x' = a, y' --=~b, #' = c, would denote a translation of the body as 

 if rigid, which is unaccompanied by relative displacement or strain. 



Let the solutions of the equations 1), which we shall term the 

 direct substitution, be 



2) 

 where 



A B x' 



. . , 



z/ etc. 



z/', etc. 



4ii ft* 



I .' C 2 j C S 



A strain represented by the equations 1) is said to be homogeneous. 

 If the accented letters denote initial positions, and the unaccented 

 letters final positions, the strain represented by equations 2) is said 

 to be inverse to the first strain. 



In virtue of equations 1) or 2) a linear relation between x, y, z 

 becomes a linear relation between x 1 , y', z\ Accordingly in a homo- 

 geneous strain a plane remains a plane, and a straight line, being 

 the intersection of two planes, remains a straight line. Finite points 

 remain finite, since the coefficients are finite, accordingly parallel 

 lines, intersecting at infinity remain parallel. Parallelograms remain 

 parallelograms (their angles being in general changed), and therefore 

 the changes of length experienced by equal parallel lines are equal, 

 and for unequal parallel lines proportional to their lengths. Thus 

 any portion of the body experiences the same change of size and 

 shape as any equal and similarly placed portion at any other part 

 of the body. This is the meaning of the term homogeneous, which 

 signifies alike all over. 



When two vectors OP of length r and OP' of length r' drawn 

 from the same origin are so related that their respective components 

 x, y, 2, x\ y\ z' are connected by the equations 1) or their equi- 

 valents 2) either vector is said to be a linear vector function of the 

 other. The properties of such linear functions are of great importance 

 in mathematical physics, and will now be taken up before their 

 application to strain. 



