Note that if P is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by 1 4 π π R − − a 4 π π ρ ρ R ′ , where R denotes the distance to the source point P and R ′ denotes the distance to the reflected point P ′. A consequence of this expression for the Green's function is the Poisson integral formula. Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values g inside the sphere is given by(Zachmanoglou & Thoe 1986, p. 228) u ( P ) = 1 4 π π a 3 ( 1 − − ρ ρ 2 a 2 ) ∫ ∫ 0 2 π π ∫ ∫ 0 π π g ( θ θ ′ , φ φ ′ ) sin ⁡ ⁡ θ θ ′ ( a 2 + ρ ρ 2 − − 2 a ρ ρ cos ⁡ ⁡ Θ Θ ) 3 2 d θ θ ′ d φ φ ′ where cos ⁡ ⁡ Θ Θ = cos ⁡ ⁡ θ θ cos ⁡ ⁡ θ θ ′ + sin ⁡ ⁡ θ θ sin ⁡ ⁡ θ θ ′ cos ⁡ ⁡ ( φ φ − − φ φ ′ ) is the cosine of the angle between (θ, φ) and (θ ′, φ ′). A simple consequence of this formula is that if u is a harmonic function, then the value of u at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.