before and after the Sinking of a Bore-hole. 789 



Filially the values of rr in the core and layer must be equal 

 over the common surface, and so 



r(3 f n-n)B-4nb-*C = m»i'-n')A + 1 *f + W ' 2ir( /0 + p') 2 & 8 

 ° ' oV ' 15(m +?i ) 



5m + n „ 72 . m — 7i 4 , ;2 , 10 . 



27r/r& 2 H -irpp'b". . (12) 



15(??i + >?) r m+ « 3 



These three equations determine A, B, and C without 

 ambiguity, and there is no difficulty, except in the length of 

 the expressions, in obtaining the solution for the general case. 



As we shall see, however, physical interest is mainly if not 

 entirely restricted to the case when the material of the core 

 is incompressible or very nearly so, and I shall thus limit 

 myself to the case when n'/m' is negligible, n' being finite. 



From (12) we see that when n'/nt is negligible A must 

 vanish. For n being finite, m' must be infinite, and so tlie 

 one term J(3m' — n')A unless A vanished would become 

 infinite, whilst the other terms in the equation would remain 

 finite. We must thus treat A as negligible, except when 

 multiplied by m f , and regard (12) as simply determining the 

 (finite) value of (3m' -n') A/3. 



Our surface equations thus reduce to two, viz. (10) and 



B + 3S-3C=-f^4 27rpp7 ' 2 , . . (11') 

 m-\-n m + n 



this being the form taken by (11) when A is negligible and 

 m' is infinite. 



The values of B and C may be at once written down from 

 (10) and (11'). Confining our attention to the case where 

 the thickness a — b — t of the crust is so small that (t/a) 2 is 

 negligible, we find 



B = 27rpa 2 {-p + 2p'(l-3t/a)} + {?>(m + n)}^ 



C = 27r/)a 5 {fp + p'(l-3^a)} + {9(m + n)}. J ' 



Substituting these values of B and C in (9) , then putting 

 r = a — h, and neglecting {h/a) 2 , we find after reduction 



r~r = —f i irpah{p + p' — 3(t/a)p'}, 



ee 



=-*^*s+s{>v-«(*/«y}- 



(14) 



But if g be the acceleration of gravity at the surface 



g= (4r/3){f»?+p'P}/<*. 

 = J?ra{/>+p'— (3t/a)p'}, 



when [t/a) 1 is neglected. 



