On the Thickness of the Crust of the Earth. 445 
inside it, its outer surface, as proved by observation, and its 
inner surface, being the first fluid layer, ex necessitate rei, are 
perpendicular to the force of gravity. The general eondition 
requisite for any surface of given specific gravity of the earth to 
be perpendicular to gravity, is contained in equation (A), which 
I here reproduce. 
ym oc I 
o=¢ ("pe sal? da oY Pas aa “lied 
In this equation, the letters signify,— 
e, the ellipticity of any layer ; 
a, the equicapacious radius of that layer ; 
p, the specific gravity of the layer ; 
a, the equicapacious radius of the outer surface of the sup- 
posed crust ; 
m, the ratio of centrifugal force to gravity at the equator. 
This equation (A) applies to the outer surface of the crust, 
and to the outer surface of the fluid nucleus, both of which are 
perpendicular to gravity. 
Let a, e denote the radius and ellipticity of the outer surface 
of the crust ; 
and iet a,, e, denote the radius and ellipticity of the outer sur- 
face of the supposed fluid nucleus: then equation (A), applied to 
these two surfaces, will become 
2(? d.a’e = 
3A, Pati 3 = 2e—m)a? | pa’, < phi) ecm ie hiner in . . (A,) 
0 
0 
and 
Aly ae Bie ey d.ae a)” ade _ma(* , 
a od 5a? J, Pda 5 Pda een iia 4a Aiba) 
The first of these equations (A,) gives Clairaut’s theorem, but 
teaches us absolutely nothing of the structure of the interior of 
the earth, except that it must be arranged in nearly spherical 
strata, each of constant density, or in some way or other equiva- 
lent to this. 
The second equation (A,) contains four definite integrals ; 
viz. 
is { pa’. 
0 
This integral extends through the whole earth, and is known, 
because the mass of the earth is known. 
a) 'm dave 
2 
Il. \ pa’, and (p Pag 
These integrals extend through the fluid nucleus, and are un- 
