M = ∫∫∫ ρ dxdydz, (1)
where the integration is to be extended throughout the volume V, may be divided into two parts by considering separately the thin shell or skin extending from the outer surface to a depth ε, within which the density and other properties of the liquid vary with the depth, and the interior portion of the liquid within which its properties are constant.
Since ε is a line of insensible magnitude compared with the dimensions of the mass of liquid and the principal radii of curvature of its surface, the volume of the shell whose surface is S and thickness ε will be Sε, and that of the interior space will be V − Sε.
If we suppose a normal ν less than ε to be drawn from the surface S into the liquid, we may divide the shell into elementary shells whose thickness is dν, in each of which the density and other properties of the liquid will be constant.
The volume of one of these shells will be Sdν. Its mass will be Sρdν. The mass of the whole shell will therefore be S∫ε0 ρdν, and that of the interior part of the liquid (V − Sε)ρ0. We thus find for the whole mass of the liquid
M = V ρ0 − S ∫ε0(ρ0 − ρ) dν. (2)
To find the potential energy we have to integrate
∫∫∫ χρ dxdydz. (3)
Substituting χρ for ρ in the process we have just gone through, we find
E = Vχ0ρ0 − S ∫ε0 (χ0ρ0 − χρ) dν. (4)