If the cylindrical multipoles are exterior, this equation becomes U = − − Q 1 2 π π ϵ ϵ ∫ ∫ d ρ ρ ρ ρ λ λ ( ρ ρ , θ θ ) ln ⁡ ⁡ ρ ρ + 1 2 π π ϵ ϵ ∑ ∑ k = 1 ∞ ∞ ∫ ∫ d θ θ d ρ ρ [ C 1 k cos ⁡ ⁡ k θ θ ρ ρ k − − 1 + S 1 k sin ⁡ ⁡ k θ θ ρ ρ k − − 1 ] λ λ ( ρ ρ , θ θ ) where Q 1 , C 1 k and S 1 k are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form U = − − Q 1 2 π π ϵ ϵ ∫ ∫ d ρ ρ ρ ρ λ λ ( ρ ρ , θ θ ) ln ⁡ ⁡ ρ ρ + 1 2 π π ϵ ϵ ∑ ∑ k = 1 ∞ ∞ k ( C 1 k I 2 k + S 1 k J 2 k ) where I 2 k and J 2 k are the interior cylindrical multipoles of the second charge density.