Relations among the ‘original’ process capability index, Cp:, and the ‘second generation’ indices, Cpk: and Cpm:, are studied in some detail, with special attention to conditions under which Cpm: is greater, or less, than Cpk:.Similar, though less extensive, analysis is applied in comparing a new index, Cpq:, with Cpm: and Cpk:. There is also a short section describing some features of the structure of Vännman's recently formulated ‘unified family’ of indices, Cp:(u, v), that includes Cp:, Cpm: and Cpk:.