or expressing dθ in terms of dƒ by (26),
σur-1ƒ dƒ dθ.
Multiplying this by m and by π(ƒ), we obtain for the work done by the attraction of this element when m is brought from an infinite distance to P1,
mσur-1 ƒΠ(ƒ) dƒdω.
Integrating with respect to ƒ from ƒ = z to ƒ = a, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the radii of curvature, we obtain for the work
∫ mσur-1 (ψ(z) − ψ(a)) dω,
and since ψ(a) is an insensible quantity we may omit it. We may also write
ur-1 = 1 + zu-1 + &c.,
since z is very small compared with u, and expressing u in terms of ω by (25), we find
| ∫2π0mσ ψ(z){ 1 + z ( | cos²ω | + | sin²ω | )} dω = 2πmσ ψ(z){ 1 + | 1 | z ( | 1 | + | 1 | )}. |
| R1 | R2 | 2 | R1 | R2 |