Substituting this equation for cos ⁡ ⁡ γ γ into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem P ℓ ℓ ( cos ⁡ ⁡ γ γ ) = 4 π π 2 ℓ ℓ + 1 ∑ ∑ m = − − ℓ ℓ ℓ ℓ Y ℓ ℓ m ( θ θ , ϕ ϕ ) Y ℓ ℓ m ∗ ∗ ( θ θ ′ , ϕ ϕ ′ ) where the Y ℓ ℓ m functions are the spherical harmonics. Substitution of this formula into the potential yields Φ Φ ( r ) = q 4 π π ε ε r ∑ ∑ ℓ ℓ = 0 ∞ ∞ ( r ′ r ) ℓ ℓ ( 4 π π 2 ℓ ℓ + 1 ) ∑ ∑ m = − − ℓ ℓ ℓ ℓ Y ℓ ℓ m ( θ θ , ϕ ϕ ) Y ℓ ℓ m ∗ ∗ ( θ θ ′ , ϕ ϕ ′ )