This paper studies two particular cases of the general 2-parameter eigenvalue problem namely

where A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.

On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

The general paraxial particle-optic properties of the hexapole field are summarized and lens formulae derived. Formulae are given for the transmission as a function of particle velocity and system geometry for geometries appropriate to the case of neutral atom beams and of slow neutron beams respectively. These formulae are used to evaluate the hexapole magnet as a velocity selector and polarizer for atomic beams and as a spin polarizer for neutron beams. Some experimental observations on potassium beams are quoted in support of the theory.