before and after tJie Sinking of a Bore-hole. 787 



In cylindrical coordinates the displacements are u along r, 

 the outward drawn perpendicular from the cylindrical axis, or 

 axis o£ z, and w parallel to this axis. The principal stresses in 



the problems treated here are rr and zz, parallel respectively 



to r and z, and <£<£ perpendicular to these two directions. 

 The corresponding principal strains are du/dr, div/dz, 

 and u/r. 



The dilatation A is given in spherical and cylindrical 

 coordinates respectively, in the cases o£ symmetry here 

 considered, bv 



A = du/dr + 2v/r, (2) 



A = dv/dr + u/r + dw/dz (3) 



Homogeneous isotropic gravitating sphere. 



§ 4. If a be the radius, p the density, and g ' gravity' at 

 the surface, then, the centre being origin, 



a pi 



a = 



< r~ — a l ~ > , 



+ n) L oni — nj 



10a[m 



rr = —gp(a 2 — r' 2 ) (5m+n)-r \lOa(m + n)\ , 

 06 = -gp{a 2 (bm + n) -r 2 {5m-3n) } -j- { 10a(m + n) }, ) ( 4 )* 



S = 2gpr 2 n-r- { oa [m + n) } , \ 



S (at surface) = 2gpan-i- {5(m + n)}~lgpa(l — 2^)/(l — tj). i 

 At points near the surface for which 

 h=a—r is small, 



we have, retaining only the lowest power of h, as first 

 approximations, 



rr=— gph$—rj)-r- {5(1— rj) }, 



d6=-S=-gpa{l-2rj) 

 In the special case when the material is incompressible, 

 i. e. n/m = 0, or rj = 0"5, 



we have throughout the mass of the sphere 

 u = 0, "\ 



?r = 66 = -gp(a 2 - r 3 )/2a, (■ .... (6) 

 S = S = 0; J 



* Cauib. Phil. Soc. Traus. vol. xiv. p. 281. 



:t-„.} ■ • <=> 



