^{page 249 note *} If this formula be compared with that given in the paper by Mr. T. B. Sprague, M.A., “On the Value of Apportionable Annuities” &c., in vol. xiii, page 3G3, of the Journal, for the value of the proportionate payment to the day of death, or, as it is there called, the correction, in the ease of an annuity payable annually, viz., it will he seen that as the correction is the only payment in that case in the year of death, the value of the total payment made in that year remains the same with the exception of small quantities involving i, i ², &c., whether the annuity be payable annually or m times a year. Supposing the latter to be the case, and that the death occurs at the expiration of the portion of the year of death, then t payments of each have been already made in that year, and the correction is . These are, together, . But if the annuity be payable annually, and T denote the correction in that case, then , since is the proportion of the year elapsed from the previous payment to the date of death. The same is true for all possible values of , or T; and if it be borne in mind that T is payable when th e death has occurred, whereas the t payments of each are made previously to the death, an explanation is afforded of the fact that the value of the total payments in the year of death in the case of an annuity payable m times a year differs from the value of the correction in th e case of an annuity payable annually by quantities involving i, i ², &c.