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The existence of a non-trivial periodic solution for the autonomous Rayleigh equation $\ddot{x}+F\left(\dot{x}\right)+g\left(x\right)=0$ is proved, assuming conditions which do not imply that $F\left(x\right)x$ has a definite sign for $\left|x\right|$ large. A similar result is obtained for the periodically forced equation $\ddot{x}+F\left(\dot{x}\right)+g\left(x\right)=e\left(t\right)$.