The orthogonality of the Legendre polynomials can be proven using the Legendre differential equation.
If 
and 
are solutions to eq332a, then
\frac{dP_n}{dx}\biggr\]+n(n+1)P_n=0\;\;\;\;\;\;\;\;351)
\frac{dP_m}{dx}\biggr\]+m(m+1)P_m=0\;\;\;\;\;\;\;\;352)
Multiplying eq351 and eq352 by and , respectively, and subtracting the results yields
\biggr\(P_m\frac{dP_n}{dx}-P_n\frac{dP_m}{dx}\biggr\)+\[n(n+1)-m(m+1)\]P_mP_n=0\;\;\;\;\;\;\;\;353)
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
\biggr\(P_m\frac{dP_n}{dx}-P_n\frac{dP_m}{dx}\biggr\)\biggr\]_{-1}^1+\[n(n+1)-m(m+1)\]\int_{-1}^1P_mP_n\;dx=0)
-m(m+1)\]\int_{-1}^1P_mP_n\;dx=0)
If 
, the factor -m(m+1)])
is not zero, and hence
