Elastic Equilibrium of Semi-Infinite Solid. 35 



At the centre of the loaded circle, since 

 Jc = 0, K = E = ^, 



o 



we get 



(^,X = ^ . J;:;^^\ /) (80) 



7^a 2{X + ii)n 



and at the periphery of the circle (>' = «)> since 



A- = 1 , E' = 1, Ji -^ log — , r,— a) K -> 0, 



we have 





The values of (il), at the centre and at the periphery of the circle 

 bear the constant ratio n-/2, and this is independent of the elastic 

 constants and radius. 



At the centre, as will be seen from tlxe formula (SO), the 

 vertical displacement varies inversely as the radius, when the same 

 amount of total pressure is applied to different circular areas, while 

 it varies directly as the radius when the pressure of the same 

 intensity is applied to different areas. The same relation liolds at 

 the edge of the circle, with regard to both the components, radial 

 and vertical, of the displacement. 



By the aid of the Legexdre table of K and Jß, we can trace 

 by a graph the general march of (i/,:)o. The next diagram is drawn 

 in this way, where the radius a is taken as unity and the pressure 

 Z7 is taken equal to 27vafJt(?< + fj.)/{^^ + '2>/j.) 



§28. To find the formuke for stress we need two more 

 integrals which can be also carried out by the same method as 

 before. AVe have, from (01), tliat 



1) This result may be obtained from (58) directly hj usjng the foruiula ./ Ji[ka) -7— =1- 

 The i-esult (80) and (81) agree with those givtn by Bodssinesq. p. 140, I.e. 



