This paper is concerned with eigenvalue problems for fourth order differential equations representable by systems of the form: $$x'' + q_{11} (t,\lambda )x + q_{12} (t,\lambda )y = 0, y'' + q_{21} (t,\lambda )x + q_{22} (t,\lambda )y = 0$$ . The boundary conditions are x(a)=y(a)=0=x(b)=y(b) or x(a)=y(a)=0=x′(a)=y′(b). Sufficient conditions for existence of least eigenvalues are given and their properties are discussed.