In former papers, I have investigated the value of an annuity on a single life, when payable half-yearly, quarterly, &c. It still remains to consider the values of annuities on the joint duration of two or more lives, or on the life of the last survivor of several lives, under similar circumstances. It will, however, be more useful to consider first the increase in the value of an annuity on a single life, when a proportionate part of the annuity is payable up to the day of death, instead of the annuity ceasing, as is generally assumed, with the payment that precedes the date of death.