before an J after the Sinking of a Bore-hole. 797 



As this method of obtaining the stresses is very artificial, I 

 may add that I have deduced directly from the second of 

 equations (37) for points in the axis of r 



? r=(T/2/0[(l-^;V-2(l-7 7 )(a' 2 + ^ + ^V 2 + ^)^]- 



Tims in the limit when z = 0, i. e. at the centre of the 

 loaded area. 



dw/dz=T(l-2ri)/2n. 



I have also succeeded in deducing from the first of equations 

 (37) as limiting values when £ = 



ot/x=/3/y=u/r=duldr=T(l-2 v )/±n. 



These values for the strains are in harmony with the values 

 given above for the stresses. 



In the case of tension or pressure over a very small area 

 it is practically immaterial whether the bounding surface is a 

 plane or a sphere of large radius. Thus in practice we can 

 regard (15) and (37) as applicable to an elastic " Earth " so 

 long as h is a very small fraction of the radius. Super- 

 posing the two stress systems, and replacing T by g p A, we 

 have a small area ira" 2 at a depth h from the surface free 

 from pressure, whilst other elements at the same depth 

 experience a pressure gph. The solution thus answers 

 closely to the conditions existing below a bore-hole of depth h. 

 At the base of the hole, at its centre, we find combining (15) 

 and (42), using the notation of (15)_, 



rr=0, 



8=0 = **^^ 



L—rj 1 



(43) 



This value of S it will be noticed hears to that in (15) the 

 ratio 



1 + v : 2. 



Applications to the Earth. 



§ 14. The first question to consider is the state ofc' matters 

 prior to the existence of the bore-hole. The Earth is spheroidal 

 and rotates about its axis of figure, circumstances which intro- 

 duce variety into the conditions in different latitudes. Results 

 such as (17), (18), and (19) -how that an elastic sphere of 



the Earth's size rotating in a day must suffer stresses from the 

 '• centrifugal forces ;; which are very considerable even for 

 material such I. The results (20) and ( 1 1 ) show, 



Phil. Mag. S. 6. Vol. 9. No. 54. June 1905. 3 G 



