We study the principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem: − Δ u ( x ) = λ g ( x ) u ( x ) , x ∈ D ; ( ∂ u / ∂ n ) ( x ) + α u ( x ) = 0 , x ∈ ∂ D , where Δ is the standard Laplace operator, D is a bounded domain with smooth boundary, g : D → ℝ is a smooth function which changes sign on D and α ∈ ℝ . We discuss the relation between α and the principal eigenvalues.