It is assumed that pure anti-symmetric vibrations of an aircraft can exist, involving fuselage torsion but excluding fuselage bending. With this assumption, which in most cases is a reasonable approximation, the eigenvalue equations for anti-symmetric vibrations of a complete aircraft are derived in a very general form. The “lumped mass” approximation to the continuous mass distribution is used and sub-matrices are associated with properties of relatively simple branches of the system. The final eigenvalue equations are expressed in terms of these sub-matrices so that in a numerical application the physical system as such is considered only in relation to properties of the simple branches. It is assumed initially that the aircraft wing and tail have flexural axes of the conventional type, but the treatment is generalised to cover swept and cranked wing aircraft.