There has been extensive work on the existence of periodic solutions for nonlinear
second order autonomous differantial equations, but little work regarding the third
order problems. The popular Poincare-Bendixon theorem applies well to the former
but not the latter (see  and ). We give a necessary condition for the existence of
periodic solutions for the third order autonomous systems, This may become useful
in further investigations. Our claims are proved and supported by certain examples.