In 1869 Mr. Sprague read before the Institute an interesting paper “On the value of Reversionary Annuities payable half-yearly, quarterly, &c., according to the conditions which prevail in practice.” This valuable paper is given in the Journal, vol. xv, page 126. The conditions which prevail in practice, as here for the first time introduced and mathematically investigated by Mr. Sprague, are the following:—For a yearly reversionary annuity on a life x after the death of another life y, the first payment of the annuity is supposed to be made just one year after the day of the death of y, provided that x be then living; and for a like annuity payable by m instalments in each year, the first payment of an instalment is supposed to be made just of a year after the day of the death of y. Also, in each case, it is further supposed that whenever x shall die, after surviving y, the annuity shall be completed up to the day of the death of x by the supplementary payment of the proportion of the annuity estimated in respect of the fractional interval that may then have elapsed since the last yearly payment or instalment was received.