The orthogonality of the Legendre polynomials can be proven using the Legendre differential equation.

If and are solutions to eq332a, then

Multiplying eq351 and eq352 by and , respectively, and subtracting the results yields

The Legendre polynomials are used to describe spherical harmonics, where and . Therefore, the orthogonality of the Legendre polynomials is analysed within the interval of . Integrating eq353 with respect to gives

If , the factor is not zero, and hence