Similarly the interior multipole expansion becomes Φ Φ ( r ) = 1 4 π π ε ε ∑ ∑ ℓ ℓ = 0 ∞ ∞ I ℓ ℓ r ℓ ℓ P ℓ ℓ ( cos ⁡ ⁡ θ θ ) where the axially symmetric interior multipole moments are defined I ℓ ℓ = d e f ∫ ∫ d r ′ ρ ρ ( r ′ ) ( r ′ ) ℓ ℓ + 1 P ℓ ℓ ( cos ⁡ ⁡ θ θ ′ ) In the limit that the charge is confined to the z -axis, we recover the interior axial multipole moments.