Theory of the Contact of Elastic Bodies. 



327 



disappears and /> 3 = becomes a cone, one of whose genera- 

 tors is shown by the broken line. This is Boussinesq'scase* 

 of pressure at a point. Inside the cone there is hoop-pressure, 

 outside hoop-tension. The angle of the cone is found to 



Fio;. 5. 



I 2 >3 



be independent of the elastic constants and to satisfy the 

 equation cos 2 + cos 6 = 1, 



giving cos0 = i(V5-l), = 51° 50'. 



In this limiting case, p 2 is everywhere negative. 



5. Lines of Principal Stress. 



The other question suggested is the limiting value 

 approached by the angle between p 1 and the radius vector 

 as r increases in any direction 6. To find this it is sufficient 

 to use the results of the simple case just referred to, in which 

 the area of contact is regarded as a point, and <f) = ¥/r. On 

 working out the strain-components in polar co-ordinates the 

 inclination yjr of the principal stress j>i to the radius vector 

 is found to be given by the equation 



tan 2^ = 2(l-2o-) sin 6 cos 6j{ (5-4(7) cos 2 6 



+ 3cos0— (l-2<r)h 



* Love, ' Elasticity,' p. 189. 



