Shales constitute about 75% of the clastic fill of sedimentary basins and have a decisive effect on fluid flow and seismic wave propagation because of their low permeability and anisotropic microstructure. The elastic stiffnesses of a shale with partially oriented clay particles is expressed in terms of the coefficients Wlmn in an expansion of the clay-particle orientation distribution function in generalized Legendre functions. Application is made to the determination of the anellipticity of shales. For transverse isotropy the anellipticity quantifies the deviation of the P wave slowness curve from an ellipse and is shown to depend on a single coefficient W400 in the expansion of the clay-particle orientation distribution function. If W400 is small, the anellipticity may be neglected, as is apparently the case for a near-surface late Tertiary shale studied by Winterstein and Paulson. Strongly aligned clay particles result in a positive value of W400 and a positive anellipticity, in agreement with the majority of field measurements. However, less well ordered shales could have a significantly positive second moment W200 but only a small positive or even negative value of W400. For such shales the anellipticity would be small or negative despite a preferred alignment of clay particles in the bedding plane. Numerical examples of clay particles distribution functions leading to zero or negative anellipticity are given.