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abscissa Actuaries Aggregate Males calculation column curve yx curves ux cx vanishes death rate derivatives differences in Table DIFFERENCES OF DEATHS differences opposite age eleven registration ENUMERATED AND GRADUATED equations expression for yx fifth difference five leading unit form leading unit form the leading function Tx Gradu Inst interpolated values interpolation formula intervals Jour June Karup leading quinquennial differences leading unit differences Mathematics and Insurance Mortality Statistics mortality table multiples of five multipliers opposite age 15 osculatory formula osculatory inter osculatory interpolation curve osculatory method persons aged POPULATION AGED Population Deaths quennial difference quinquennial age groups quinquennial values registration area registration of deaths relation successive interpolation curves TABLE BY OSCULATORY Table VII taken from Table THIRD NUMERICAL ILLUSTRATION tion English Twelfth Census Tx and lx Tx+w unit differences opposite United States Census University of Michigan ux and vx ux is defined whence York City