168 a] ELONGATION QUADRIC. 443 



64) (s x ) x* + (s 9 S)y 2 + ($ y S)z*+ 2g x yz + %g y xz + 2g g xy = 0, 



which may be called a cone of equal elongation S, of which the cone 

 of no elongation is a particular case. 



Let us form the elongation quadrics for expansions and shears. 

 If the slides vanish we have 



65) x = SxX* + s,y 2 + s,s* = 1, 

 and for a simple stretch in the X- direction 



66) s x x* = 1, 



the elongation quadric breaks up into the two parallel planes, 



l^SxX 1=0 and y ' s x x +1=0, 

 at distances + from the origin. 



Since for any line making the angle # with the X-axis we have 



r = = 9 



ys x cos # 



the stretch is given by 



67) s r = -g = s x cos 2 #. 



The cone of no elongation is therefore the plane & = ^ parallel to 



the above pair of planes. In equation 65) if s x) s y , s z are of the same 

 sign the quadric is an ellipsoid and the cone of no expansion is 

 imaginary. If one s has a sign different from that of the others we 

 have two hyperboloids and the cone of no expansion is real and 

 separates the stretched from the squeezed lines. 

 In the general shear s x = Sy = s z = we have 



68) x = 2 (g x yz + g y ex + gxy) = 1, 

 and the cone of no elongation 



69) g x yz 4 g y sx + g e xy = 0, 



contains the three coordinate -axes as generators. These are therefore 

 unstretched. In a simple shear parallel to the XY- plane we have 



70) % = 2g z xy = 1 



which represents equilateral hyperbolic cylinders with axes bisecting 

 the angles between the x and y axes. The cone of no elongation, 

 xy = 0, breaks up into two coordinate -planes, x = and y = 0. 

 These two planes are undistorted, and are the planes of circular 

 section of the strain -ellipsoid. 



