before and after the Sinking of a Bore-hole. 791 



latitude, 

 ff = ti6 = a> 2 P a 2 (?>-6r)-5r ] 2 )^{5(l- V ){l + 5y)\, 



»- 5(l J5? + W {3-6^ + 5cob'X(W) ]. 



(17) 



5(1-^(7 + 51?) 



The value of 66 is thus constant over the surface. Its sign 

 depends on that of 3 — 6rj — brj 1 . This expression vanishes 

 when 77-"3798, or »/w='2404. It also vanishes when 

 »i//i = *2404, but this latter value is physically impossible. 



Thus 66 is positive or negative, i.e. a tension or a pressure, 



according as rj is less or greater than 0'38. (jxj) is always 



positive in the equator ; at the poles it is equal to 66. 



When </></> is positive and 66 negative (which implies rj 



being >*3798) <fxp— 66 is the maximum stress difference, 

 and the greatest value occurs in the equator where 



S=«> 2 (I+i ? )/(7 + 5i ? ) (18) 



This increases slightly with ??, varying from 0'155&) 2 joa 2 

 when ff= '3798 to 0-lZ>$a) 2 pa 2 when ^ = 0*5. 



Supposing the earth an incompressible sphere in which 

 p = 5o, a = 3963 miles, and (o 2 a=g/2d i d, we find for the 

 maximum surface value of S 



S = 12'0 tons weight per square inch. . . (19) 



Except for incompressible material, the principal terms 



depending on co 2 in tt and <j>j> are small compared with the 

 principal gravitational terms ; and when the material is incom- 

 pressible the secondary terms depending on co 2 which contain 

 u 2 — c 2 are small compared with the secondary gravitational 

 terms. Thus we may in general neglect the secondary terms in 

 co 2 for a first approximation. We may also neglect the differ- 

 ences between a, c, and p in all terms depending on co or 

 containing a 2 — c 2 as a factor, and may replace 47rGpa/3 by g y 

 where g is the mean value of gravity over the surface. 



Doing so, we find as first approximations to the principal 

 stresses in the surface of an incompressible nearly spherical 

 Bj heroid 



wi = 0, 1 



tt= T9 epa \l~a —j) ' • (20) 



— 1 (4 a — c (o 2 a\( r 2 \ 



*+ = i<j' ,pa {r,— -t)\ °jn 



