^{page 161 note *} The truth of the several formulæ given in this paper will be obvious on inspection, by bearing this in mind:—Thus, since the purchaser must always have interest on his capital for the year in which the death happens, the advance for £1 must not exceed v or 1–d; in this case, too, he must deduct p, the first year's premium; and since d is the interest upon 1–d, the annuity he should get for 1–d–p is (d+p); so that £1 annuity is worth or ; that is to say, the sum to be assured, less the first year's premium and the last year's interest.

^{page 162 note *} Here again the advance for £1 reversion is v or 1–d, in addition to which, the interest upon 1 – d during the life of A has to be provided for. The sum to be given is therefore 1–d – dA, or 1–d (1 +A), or v–dA.

^{page 162 note †} Here the purchaser, for the same reason as before, must limit his advance for £1 to v; from which he has to deduct the first year's premium p, and the cost of an annuity to provide interest and premium—that is to say, of (d + p)—during the joint-lives. Hence the difference will be 1– d– p – (d + p)AB, or 1 – (d + p) (1 + AB). He will thus receive interest, and have the means of paying the premium, while both lives are in being; and should A die first, he will get £1 = v + d, the total outlay, with interest upon it for the last year, from the estate; or should B die first, he will get it from the Assurance Office.

^{page 163 note *} In this case, as in the foregoing ones, the advance must be limited to 1 –d; from which have to be deducted the first year's premium, and the cost of an annuity equivalent to the annual premium and interest, whilst both A and B are living. The sum to be given therefore for a contingent reversionary annuity of p + d, under these circumstances, woul d be 1–d–p –(p + d) AB, or 1 – (p + d) (1 + AB). The purchaser would then receive his annuity of (p + d) so long as B is alive,—viz., from the Annuity Company whilst A is living, and from the estate should he die; and at B's death he would recover his capital as before, with interest for the year in which his death occurred. But if p + d annuity is worth 1 – (p + d) (1 +AB), £1 annuity is worth , or , as above given. In this expression, represents the sum to be assured; and in strict analogy with the former cases, is identical with that to be given for such an annuity, after deducting the first year's premium, the last year's interest, an d the sum required to provide both so long as the joint-lives are in being.

^{page 165 note *} These are what may be called practicable rates. To assume ones less costly, or much less costly, would be to build on a false foundation!