The solutions of the Helmholtz vector equation obey the following orthogonality relations: ∫ ∫ 0 2 π π ∫ ∫ 0 π π L o e m n ⋅ ⋅ L o e m n sin ⁡ ⁡ ϑ ϑ d ϑ ϑ d φ φ = ( 1 + δ δ m , 0 ) 2 π π ( 2 n + 1 ) 2 ( n + m ) ! ( n − − m ) ! k 2 { n [ z n − − 1 ( k r ) ] 2 + ( n + 1 ) [ z n + 1 ( k r ) ] 2 } ∫ ∫ 0 2 π π ∫ ∫ 0 π π M o e m n ⋅ ⋅ M o e m n sin ⁡ ⁡ ϑ ϑ d ϑ ϑ d φ φ = ( 1 + δ δ m , 0 ) 2 π π 2 n + 1 ( n + m ) ! ( n − − m ) ! n ( n + 1 ) [ z n ( k r ) ] 2 ∫ ∫ 0 2 π π ∫ ∫ 0 π π N o e m n ⋅ ⋅ N o e m n sin ⁡ ⁡ ϑ ϑ d ϑ ϑ d φ φ = ( 1 + δ δ m , 0 ) 2 π π ( 2 n + 1 ) 2 ( n + m ) ! ( n − − m ) ! n ( n + 1 ) { ( n + 1 ) [ z n − − 1 ( k r ) ] 2 + n [ z n + 1 ( k r ) ] 2 } ∫ ∫ 0 π π ∫ ∫ 0 2 π π L o e m n ⋅ ⋅ N o e m n sin ⁡ ⁡ ϑ ϑ d ϑ ϑ d φ φ = ( 1 + δ δ m , 0 ) 2 π π ( 2 n + 1 ) 2 ( n + m ) ! ( n − − m ) ! n ( n + 1 ) k { [ z n − − 1 ( k r ) ] 2 − − [ z n + 1 ( k r ) ] 2 }