We have introduced an oxygen atom and a carbon-carbon double bond with atrans-configuration (E) into the terminal alkyl chain attached to the cyclohexyl ring of a variety of two- and three-ring nematic liquid crystals of positive dielectric anisotropy. The new polar two-ring methyl (E)-allyl ethers often possess low melting points, but are not mesomorphic in general. The related three-ring methyl (E)-allyl ethers exhibit high clearing points and wide nematic ranges. Comparisons with the corresponding derivatives incorporating either just an oxygen atom or just a carbon-carbon double bond in the same position indicate that synergetic effects lead to broader nematic phases than would otherwise have been expected. This is partially due to the low smectic transition temperatures observed for the methyl (E)-allyl ethers. Selected physical properties of three binary mixtures of a weakly polar standard nematic liquid crystal and three difluoro-substituted (polar) liquid crystals (including the new ethers) differing only in the nature of the terminal chain show that, although some of the methyl (E)-allyl ethers exhibit longer switch-off times in TN cells than those of analogous liquid crystals incorporating either a methyl propyl ether or a 1-(E)-propenyl chain instead of the methyl (E)-allyl ether chain, they are still useful components for nematic mixtures, especially where a wide temperature range is required.