Consider a smooth d -dimensional manifold A embedded in R p and denote its surface measure by σ σ . We assume σ σ ( A ) > 0 . Assume s ⩾ ⩾ d As before, for every N ⩾ ⩾ 2 fix an N -point s -equilibrium configuration ω ω N ∗ ∗ = { x 1 , N , … … , x N , N } and set μ μ N := 1 N ∑ ∑ i = 1 , … … , N δ δ x i , N . Then, in the sense of weak convergence of measures, μ μ N → → ∗ ∗ μ μ , where μ μ ( B ) = σ σ ( A ∩ ∩ B ) / σ σ ( A ) . If H d is the d -dimensional Hausdorff measure normalized so that H d ( [ 0 , 1 ] d ) = 1 , then lim N → → ∞ ∞ E s ( A , N ) N 1 + s / d = 2 s α α d − − s / d ⋅ ⋅ C s , d ( H d ( A ) ) s / d , where α α d = π π d / 2 / Γ Γ ( 1 + d / 2 ) is the volume of a d-ball.