the Ea7'lh and the Geological Changes of its Suiface, 381 

 M\i io iij; • cJiJfJqiimyn 



<ii* :>^»H«^^ « (-/ (^^ «i"^ Q + /3^ cos^ @ ) ± G) ^3^ W a9no. 

 l'«9iIT .lOJcup.V (a-^sin^® + /S^cos^©) "ib'no)ej|il) 



^d b9niQjjJo_i> jj (ii^ (flt^ sin^ Q + jS^cos^ 0) ± G) ^npom 9'" 

 ^ - -v/(«2sin2 + /32cos2 0) ' noljBupa arfj 

 A and B representing the general values of the polar and 

 equatorial semi-axes of the shell corresponding to the ZQO£io£ 

 the shell whose thickness is T. /boi yJi^ 



When investigating the moment of inertia of the shell rot 

 asny zone, it will be found most convenient to consider the 

 centres of gravity of the parts of the shell, or the surface pass- 

 ing through the centres of gravity of all its parts* as eqtii- 

 distant from its interior surface.'^ * ^^ '^^ ,Y>l^nf.<- -^, .o 



No matter how irregular any zone of theshell may bcj itsf 

 thickness can be considered without much chance of error, as 

 the distance between two imaginary surfaces each of which, 

 is equidistant from the surface passing through the centres of 

 gravity of all the parts of the shell. Let the internal surface, 

 of the shell be at the mean depth of the sea, the position of its 

 external surface being determined by the distances of the cen- 

 tres of gravity of its parts. Let the entire shell be supposec}-; 

 to consist of an infinite number of pyramidal frusta, which if 

 prolonged would form pyramids meeting at the earth's centre. 

 Tile height and dimensions of the base of each frustum being 

 infinitely small compared to the height of its entire pyramid, 

 we can without sensible error consider it as a parallelopip^d. 

 Let H represent the mean height of the land above the level 

 of the sea, and D the mean depth of the ocean. The height 

 ; of a parallelopiped whose external surface is on dry land is 

 'ithen H + D, and the distance of its centre of gravity frorai= 



each of the 'surfaces of the shell - (H + D). Similarly,- D 



represents the distance of the centre of gravity of a parallelo- 

 piped whose upper surface forms a part of the surface of the 

 ocean, from the interior surface of the shell. Let As repre-, 

 sent an indefinitely small portion of the surface of the land, 

 A s' a similar portion of the surface of the sea, Sj the density of 

 the land, and Sg the density of the ocean. The equation^fjMj 

 the value of Cj, the thickness of the shell, will be — rr" i 



HS(H + D)As8t . P(§2SA5^ + SiSA5) 



: + D) ^2 A s+'Sgt) XK7} "^ 2 {2 As + 2 As) 

 Hs(D + H)S^ D(8, sM-Si s) 



~ {S,(H + D)s + 82Ds'}"^ 2(s' + s) * • • • ^'""^ 

 In an examination of the equilibrium of the degrading and 

 elevatory forces on the surface of the earth, it is indiffierent 



