^{page 189 note *} This remark is applicable only to Annuities Certain, as I find that Mr. Gray himself, in his work on Life Contingencies, has treated upon Life Annuities the successive payments of which are the several orders of figurate numbers. The connection between the values of Annuities of two successive orders is shewn by Mr. Gray to be expressed by the equation , where denotes the value of an annuity of the mth order of figurate numbers, on a life aged x. In Mr. G.'s arrangement the payment due at the end of the first year is always unity.

^{page 195 note *} Since writing the above Mr. Adler has shewn me a presentation copy of Baily's work, which belonged to the late Mr. Gompertz, in which the whole of the investigation relative to the amounts of increasing annuities is cancelled and the following note, in Baily's own handwriting, appended: “The whole of this investigation is erroneous” in principle,—it was written in haste whilst the proof was waiting. (Signed) F. B.” This explains the circumstance alluded to in the text.

Mr. Adler has also pointed out to me that the materials of Baily's two chapters are to be found in the second volume of Dodson's Mathematical Repository, to which work, indeed, Baily refers his readers for further information on the subject.

^{page 198 note *} From this it is seen that the value of the increasing perpetuity is equal to the value of a fixed perpetuity of £1. calculated upon the assumption that the rate of interest is diminished by the rate of increase in the payments. Since the text was written I have received a communication from Mr. H. Mountcastle, of the Northern Assurance Office, pointing out the same result. The proposition is by no means self-evident, and it is not true for an annuity for a limited term.