170, 171, 171 a] GEOMETRY OF STRESS. 451 



If P , P 2 , P 3 are all of the same sign the quadrics cp and 9?' 

 are ellipsoids. If they are positive we must take the positive sign 

 with R?, and the normal stress on every plane is a traction. If they 

 are negative, we must take the negative sign, and the normal stress 

 is always a pressure. If one of the P's has a different sign from 

 the two others, we use both signs and have pairs of conjugate 

 hyperboloids. In this case for directions parallel to the generators 

 of the asymptotic cone cp = to the stress quadric, we have r infinite 

 and F nn = 0. Accordingly for stress -planes perpendicular to these 

 generators, the normal stress vanishes or the stress is a shearing 

 stress. These planes envelop a cone called Lame's shear -cone, which 

 divides the directions for which the normal stress is a traction from 

 those for which it is a pressure. 



In the reciprocal quadric qp', when the radius vector is infinite, 

 it lies in its conjugate plane, the stress -plane. But the radius vector 

 to this quadric has the direction of the stress -vector, so that the 

 shear -cone is the asymptotic cone to this quadric <p r = 0. If we 

 construct the ellipsoid 



we have by 22) 



102) T=7 = f 



or the strain on a plane perpendicular to any radius vector is inversely 

 proportional to that radius vector (it does not lie in the direction 

 of the radius vector). This ellipsoid is called Cauchy's stress -ellipsoid 

 and its axes are proportional to the squares of the stress quadric 

 cp = + JR 2 . The reciprocal ellipsoid 



103) *' = j + Jj + jh = S 2 , 



has the property, since by 16) F n = ^-> that the stress -vector for any 



plane is directly proportional to the radius vector in its own direction. 

 This ellipsoid is called Lame's stress -ellipsoid, or ellipsoid of elasticity. 



171 a. Simple Stresses. A simple stress is one that contains 

 but a single constant in its specification. These are: 



1. Uniform traction or pressure. 



P~P ~P T> 



i = -L a = J-n - JC . 



All the quadrics are spheres and every stress is normal to its plane 

 and of invariable amount P. Such a stress is physically realized by 

 a body subjected to hydrostatic pressure. 



29* 



