page 98 note * Mr. Sprague has since gone at great length into the subject of annuities, and in confirming the accuracy of these results has been most liberal in his allusions to the decided improvement effected by my formulæ. It is well that Mr. Sprague has been induced to follow me in this particular investigation, as his methods are so essentially different from mine and afford such a scope for Ms acute mathematical talent, that his paper must be esteemed as a most valuable contribution to the Journal.

page 100 note * It has been pointed out to me by Mr. Merrifield that Legendre employs this formula in evaluating certain elliptic integrals in his valuable work on that subject. The formula is ascribed to Euler, and is also used by Laplace and others in the Theory of Probabilities.

page 105 note * Otherwise, if p denote the probability of surviving the interval x, then by (3),

And, since –dp is the proportion of deaths or probability of decease during the instent dx,

Also, by (2),

And, after deducting the advance payment

page 108 note * The value of a life annuity, when it is to be completed by the payment of a proportionate part up to the day of death, was first investigated in Mr. Sprague's paper, before referred to. It may be added that the formulæ (11) and (14) I have deduced for Complete Annuities are identical with his results.

Note.—The fundamental formulæ for assurances are comprehended in three types. The first of these is applicable generally, and may be written in an abbreviated form, by separating the symbols of operation, thus: which may be regarded as the radix of the formulæ (27), (31), (32), (40), and (41). The other two forms, which appertain exclusively to absolute assurances, are and they are respectively exhibited in (28), (33), (42) and (29), (34), (43). There is yet another form, to which special reference has not been made, viz which for calculation may be practically represented by