Now, to come to the case of a heterogeneous fluid, we shall assume that its surfaces of equal density are spheroids, concentric and having a common axis of rotation, and that the ellipticity of these surfaces varies from the centre to the outer surface, the density also varying. In other words, the body is composed of homogeneous spheroidal shells of variable density and ellipticity. On this supposition we shall express the attraction of the mass upon a particle in its interior, and then, taking into account the centrifugal force, form the equation expressing the condition of fluid equilibrium. The attraction of the homogeneous spheroid x² + y² + z²(1 + 2e) = c²(1 + 2e), where e is the ellipticity (of which the square is neglected), on an internal particle, whose co-ordinates are x = f, y = 0, z = h, has for its x and z components
X′ = −4⁄3πk²ρf(1 − 2⁄5e), Z′ = −4⁄3πk²ρh(1 + 4⁄5e),
the Y component being of course zero. Hence we infer that the attraction of a shell whose inner surface has an ellipticity e, and its outer surface an ellipticity e + de, the density being ρ, is expressed by
dX′ = 4⁄3 · 2⁄5πk²ρf de, dZ′ = −4⁄3 · 4⁄5πk²ρh de.
To apply this to our heterogeneous spheroid; if we put c1 for the semiaxis of that surface of equal density on which is situated the attracted point P, and c0 for the semiaxis of the outer surface, the attraction of that portion of the body which is exterior to P, namely, of all the shells which enclose P, has for components
| X0 = 8⁄15πk²f ∫c0c1 ρ | de | dc, Z0 = 16⁄15πk²h ∫c0c1 ρ | de | dc, |
| dc | dc |
both e and ρ being functions of c. Again the attraction of a homogeneous spheroid of density ρ on an external point f, h has the components
X″ = −4⁄3πk²ρfr−3 {c³(1 + 2e) − λec5},
Z″ = −4⁄3πk²ρhr−3 {c³(1 + 2e) − λ′ec5},