The disintegration of a compound thread of fluid can be utilized in a wide variety of applications including the production of compound droplets or capsules. In this paper we investigate the linear instability of a compound inviscid liquid jet falling under gravity in a surrounding gas with respect to nonaxisymmetric waves. We derive an analytical expression for the dispersion relation, which takes into account the non-uniform nature of the jet, and which we then solve numerically. Particular attention is paid to investigating the effects of the liquid-to-gas density ratio on the growth and development of different wave modes as well as the influence of gravity. Our results show that there exists some nonaxisymmetric long wavelength disturbances that are more unstable than their asymmetric equivalents and that the influence of gravity can alter the behavior of these modes.