before and after the Sinking of a Bore-hole. 795 



The stresses and stress-difference are given by 

 ^ =-/>a 3 (l-a' 3 /r 3 )/(a 3 -a' 3 ), 



09 = -paS{l+a'y-2r*)/(a*-a'Z), f 



S = 3 w <aa') 3 --{r 3 (a 3 -a' 3 )}, 

 S (at inner surface) = |pa 3 /(a 3 — a' 3 ). 

 When a' /a is very small a closely approximate value is 



S = S>/2 . (36) 



§ 13. Material bounded by an infinite plane, — on one side 

 of which it extends to infinity, — acted on by a surface-tension 

 uniformly distributed over a circular area of radius a!. 



For clearness suppose the material to be on the lower side 

 of the plane z = 0, the tension T, per unit area, acting vertically 

 upwards. Let z be measured positively downwards from an 

 origin at the centre of the stressed area, and let R denote the 

 distance between an element d<r of the stressed area and a 

 point /•, z in the solid r being the perpendicular on the axis 

 of z. 



The solution, as obtained by Boussinesq and Oerruti *, is 



u = (T/4™) j r { |j JW + (1 - 2,) jj log (_- + B)4r } , j 

 H-CE/te*) { J f) Rrfo-(3-2,)Jj i<?<r} , } . (37J 



A=-(T/2^)(l-2,)iJjj'irf<r. 



The integrals extend over the whole of the stressed area, 

 t. e. over the area enclosed by the circle r = a' '. 



The above integrals in their general form are somewhat 

 unmanageable. At a point whose distance from the stressed 

 area is such that a'/R is always small, closely approximate 

 values for the displacements are 



_ Ta'*rjz (l-&y) l ] 

 hdi 1R 2 R+5 J ' I 

 Ta' 2 f/~\ 2 1 I ' ' ' ^ 8 ^ 



where R now denotes x /r- + z'-. 



Along a given radius the above displacements vary 

 inversely as the distance from the origin, i. e. from the 



* T-jdhiinter and Pearson's ' History of Elasticity/ vol. ii. art. 1492, 

 equations (ix.). 



