∫∞ƒ ƒΠ(ƒ) dƒ = ψ(z), (24)
then 2πmσψ(z) will represent—(1) The work done by the attractive force while a particle m is brought from an infinite distance to a distance z from an infinitely thin stratum of the substance whose mass per unit of area is σ; (2) The attraction of a particle m placed at a distance z from the plane surface of an infinite solid whose density is σ.
| Fig. 2 |
Let us examine the case in which the particle m is placed at a distance z from a curved stratum of the substance, whose principal radii of curvature are R1 and R2. Let P (fig. 2) be the particle and PB a normal to the surface. Let the plane of the paper be a normal section of the surface of the stratum at the point B, making an angle ω with the section whose radius of curvature is R1. Then if O is the centre of curvature in the plane of the paper, and BO = u,
| 1 | = | cos²ω | + | sin²ω | . (25) |
| u | R1 | R2 |
Let
POQ = θ, PO = r, PQ = ƒ, BP = z,
ƒ² = u² + r² − 2ur cos θ. (26)
The element of the stratum at Q may be expressed by
σu² sin θ dθdω,