Sphingomyelin and cholesterol are of interest to biologists because they interact to form condensed structures said to be responsible for a variety of functions that membranes perform. Synchrotron X-ray diffraction methods have been used to investigate the structure of bilayers of D-erythro palmitoyl-sphingomyelin and complexes formed by palmitoyl and egg-sphingomyelin with cholesterol in aqueous multibilayer dispersions. D-erythro palmitoyl sphingomyelin bilayers exist in two conformers that are distinguished by their lamellar repeat spacing, bilayer thickness, and polar group hydration. The distinction is attributed to hydrogen bonding to water or to intermolecular hydrogen bonds that are disrupted by the formation of ripple structure. The coexisting bilayer structures of pure palmitoyl sphingomyelin are observed in the presence of cholesterol-rich bilayers that are characterized by different bilayer parameters. The presence of cholesterol preferentially affects the conformer of D-erythro sphingomyelin with thicker, more hydrated bilayers. Coexisting bilayers of sphingomyelin and complexes with cholesterol are in register and remain coupled at temperatures at least up to 50 degrees C. Cholesterol forms a complex of 1.8 mols of sphingomyelin per cholesterol at 37 degrees C that coexists with bilayers of pure sphingomyelin up to 50 degrees C. Redistribution of the two lipids takes place on cooling below the fluid- to gel-phase transition temperature, resulting in the withdrawal of sphingomyelin into gel phase and the formation of coexisting bilayers of equimolar proportions of the two lipids. Cholesterol-rich bilayers fit a stripe model at temperatures less than 37 degrees C characterized by alternating rows of sphingomyelin and cholesterol molecules. A quasicrystalline array models the arrangement at higher temperatures in which each cholesterol molecule is surrounded by seven hydrocarbon chains, each of which is in contact with two cholesterol molecules. The thickness of bilayer complexes of sphingomyelin and cholesterol is less than that of coexisting bilayers of pure sphingomyelin. The implications for protein sorting theories based on bilayer thickness are discussed.