This then expresses the work done by the attractive forces when a particle m is brought from an infinite distance to the point P at a distance z from a stratum whose surface-density is σ, and whose principal radii of curvature are R1 and R2.
To find the work done when m is brought to the point P in the neighbourhood of a solid body, the density of which is a function of the depth ν below the surface, we have only to write instead of σ ρdz, and to integrate
| 2π m∫∞zρψ(z) dz + πm ( | 1 | + | 1 | )∫∞zρzψ(z) dz, |
| R1 | R2 |
where, in general, we must suppose ρ a function of z. This expression, when integrated, gives (1) the work done on a particle m while it is brought from an infinite distance to the point P, or (2) the attraction on a long slender column normal to the surface and terminating at P, the mass of unit of length of the column being m. In the form of the theory given by Laplace, the density of the liquid was supposed to be uniform. Hence if we write
K = 2π ∫∞0 ψ(z) dz, H = 2π ∫∞0 zψ(z) dz,
the pressure of a column of the fluid itself terminating at the surface will be
ρ² {K + ½H (1/R1 + 1/R2) },
and the work done by the attractive forces when a particle m is brought to the surface of the fluid from an infinite distance will be
mρ {K + ½H (1/R1 + 1/R2) }.
If we write