{» 



Earth resulting from Secular Cooling. 135 



coefficient of dilatation at the surface, and e + e'v that at depth: 

 ai Then, using Lord Kelvin's well-known solution, expression 

 (1) becomes 



— x)jx dxLl s/( 7r )Jo J2 N /(7r/c) ^ J 



where a=2*/ (tct), and infinity is substituted for c in the 

 upper limit, Lord Kelvin's solution being adapted to the case 

 of a sphere of infinite radius. 



The shell at depth x will be stretched, unstrained or 

 crushed, according as expression (2) is positive, zero or 

 negative. 



Depth of the Surface of No Strain. — Putting x=ay and 

 6 n/ ( w ')/2"W==;8, we obtain, after division by irrelevant factors, 

 the following equation for determining the depth of the 

 surface of no strain, 



f"(<i-ay) s [y»-* , .+ 08+ [**-*•«&) (l-^y^]^ 



= JJe-^) , [jw-* , + (£+ JV-*<fa)(i- vy-**]<*y- ( 3 ) 



Since y is a proper fraction, this equation, omitting un- 

 important terms, becomes 



+/(fca 2 -tc 3 + |/3c 2 a) +<y 5 (L6 c 2 a + i/3c 3 ) +.. .... (4) 



As a first approximation, which is sufficient for most 

 purposes, we have the equation 



8qy(y + 0) = 3a(V2? + 40) (5) 



Putting j3 equal to infinity in the latter, we get 

 y = 3a/2c, 



or x=6Kt/c, (6) 



which gives the depth of the surface of no strain on the 



