790 Dr. Chree on the Stresses in the Earth's Crust 

 Thus we have in the crust : 



rr z= -gph, *) 



I 



06=-g P h(m-n)/(m + n)=-c, P h v l(l- v ), ( 

 S = gph(l-2 v )l(l- n ). 



§ 6. Slightly spheroidal, homogeneous, gravitating and 

 rotating earth. 



The surface values of the stresses, which are by no means 

 very complicated, will suffice for our immediate object. 



Let 2a, 2c represent the equatorial and polar diameters, 

 r the perpendicular from any point on the polar axis, p the 

 perpendicular from the centre on the tangent plane, p the 

 density, G the gravitational force between two unit masses 

 at unit distance, and co the angular velocity. 



At the surface the principal stresses are nn along the nor- 

 mal, tt along the tangent in the meridian plane, and <j><f> 

 perpendicular to the meridian plane. 



Neglecting terms of order, (l — c 2 /a 2 ) 2 we find* 



nn — 



Tt 



4»G P V f/ n _ 9 s, a*-<n-r,-4n >\ 

 35(l-,oV ' « 2 7 + 5/, J 



+ fP^Py { (7 + 5,X3-6,-5^) + 4^(l + .)(6-5,-5^) } , 

 o(L — r))(< +on) { a 2 J 



n 15(1-77)1 « 2 \ a 2 ) 7 + 5,7 J 



+ g,i T^ , 2 r(7 + 5>?X3-6,-5 ? 7 2 ) + 4^- 2 (l+r7)(6-5,-5, 2 

 5(1 — n)Ci +5*7) L a 



+ ^|5(7 + 5^)(l-r7 2 )-^ 2 (30-^-82r / 2 -4V)}] 



When c = a, ^irbGa/d = gthe gravitational acceleration at 

 the surface. Thus in a perfect sphere the contributions from 



the gravitational terms to tt and (fxj> are equal, and agree 

 with the value — gpa{\ — 2??)-f-{5(l — r})} given in (5) of § 4 

 for 66; they vanish when the material is incompressible. 



The contributions from the " centrifugal force" terms to 

 the surface stresses are in a perfect sphere, X denoting the 



* Roy. Soc. Proc. vol. lviii. eqns. (15), p. 43, and eqns. (23) to (26), 

 pp. 44 and 45. 



