Here Y ℓ is called a spherical harmonic function of degree ℓ and order m, P ℓ is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π /2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2 π. For a fixed integer ℓ, every solution Y (θ, φ) of the eigenvalue problem r 2 ∇ ∇ 2 Y = − − ℓ ℓ ( ℓ ℓ + 1 ) Y is a linear combination of Y ℓ. In fact, for any such solution, r Y (θ, φ) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2 ℓ + 1 linearly independent such polynomials.