^{page 288 note 1} It should perhaps be noted that, in following the plan adopted by Mr. Hardy of combining the values of e _[x−5]+5 and e _[x−10]+10 for ten consecutive values of x, and talcing the average of such ten values, the vitality of the lives is probably somewhat overstated (as compared with expectations taken at an individual age) by the inclusion of such cases of re-selection as fell within the period of ten years. Thus the successive values of e _[x−5]+5, in the combined group x=55 to 64, include all entry ages 50 to 59 inclusive, and if annuity contracts were effected on the same life (for instance) at entry ages 50, 53, and 56, the value of the average expectation deduced from the group would be enhanced, by any superior vitality included in respect of such cases of re-selection.

^{page 293 note 1} The explanation of the methods followed, and of the principles upon which they are based, has been Somewhat amplified, at the suggestion of Dr. A. E. Sprague.

^{page 296 note 1} See Appendix, p. 311.

^{page 296 note 2} I had not remembered, until I turned up this masterly review for purposes of the above reference, that the author, in discussing the graduation of the Annuity Experience, had suggested (i) (p. 360) that an ultimate Table; deduced from select data after five years, might possibly have been substituted for Mr. Hardy's hypothetical Table; (ii) (p 362) that the possibilities of the special method suggested by Mr. Hardy for the calculation of joint annuity values on two lives, of which one at least was female, might usefully be further developed later on. As these two points practically cover the ground of the present Paper, the review in question may perhaps be regarded (although quite unconsciously, so far as I am concerned) as its fons et origo.

^{page 303 note 1} The Select and Ultimate annuity values on two joint lives of equal age have since been computed at 5 per cent, interest, and are given in Tables XIX and XX of the Appendix.

^{page 311 note 1} This relation shows at once that the probabilities and annuity values can be equated, with variation both of the age and the rate of interest; and the value of the constant α could at once be deduced by equating the functions and It has, however, been preferred to set out, for the benefit of students, the fuller demonstration of the relations between the male table, and each series of the female table, as throwing useful light on the functions and constants which enter into their construction respectively.

^{page 322 note 1} In order to save space, the quantities (E_x−½θ_x)c^x+½ and Σ(E_x−½θ_x)c^x+½ for former quantity, and of its continuous summation, between the limits x=50 and summation, and of continuous summation, are precisely as set out in columns (2) each value of x, are not included in the above Table. The totals of the values of the x=90, come out at 23,401,530 and 330,348,534 respectively. The processes of and (3) above.

^{page 345 note *} Effect has been given to this suggestion (see p. 293).