404 RICE 



ART. K 



I 



structed, which are represented, in the "local" coordinates 

 ^', r\ , f ', by the equation 



where fc is a constant which has a definite value for each member 

 of the family. One member of this family will pass through Q' 

 and, if we recall the statements made concerning quadric sur- 

 faces in the author's Mathematical Note (this volume. Article 

 B, p. 15), it will be seen by reference to (6) that the dif- 

 ferential displacement Q'R" of the point Q! is normal to 

 this surface at this point. The result of this will be that 

 points originally on a straight line will still lie on a straight 

 line after the strain. (The expressions in (6) are linear in ^ , t]' , 

 f '.) But in general the angle between two lines will be altered 

 in value; in particular two lines at right angles to each other 

 before the strain will not be at right angles after it. However, 

 there is an exception to this general statement. There are three 

 mutually orthogonal directions and any lines which are parallel 

 to these before the strain remain at right angles to each other 

 after the strain. These directions are in fact the directions of 

 the three principal axes of the quadric surface; for if Q! is on one 

 of these, then, since Q'W is normal to the surface at Q', R" is 

 on the axis too, and the lines F'Q! and F'R" are coincident. 

 But by construction F"Q," is parallel to P'R"; therefore it is 

 parallel to P'Q'. Hence the three principal axes are displaced 

 into three lines parallel to them respectively, and so are at right 

 angles to each other as before. 



To prove this we apparently had to restrict our reasoning by 

 assuming that 623 = ez2, etc. We can remove this restriction 

 however and still arrive at the same result. To show this we 

 must resort to a simple artifice. Take the first expression in 

 (5), and treat it thus: 



{en - 1) r + e,W + eisf ' = (eu - D ^' + 



612 + 621 



V 



I ^31 + ei3 , 612 — 621 , 631 — en , 



