When the required annuity is payable during the joint continuance of s lives, the value expressed in terms of Gompertz's function is

where x is the youngest age and (d, d', d" denoting the differences between x and the other ages respectively) g' is determined by the equation

Example II. Required the value of a continuous annuity on two joint lives (aged 27 and 37 respectively) the rate of interest being 4 per-cent, and the constants of mortality being

Here log2=log2+log(1+q1o)=4·4075 +·5575

=4.9650

The corresponding annuity is 14-276. The value of the same annuity deduced independently from a private mortality table (constructed from the constants above given) is 14.275.

It will be observed that the trouble involved in the process of determining the required values arises solely from the limited.

extent of the table, necessitating the use of two orders of differences, and may be entirely obviated by the construction of a table on a sufficiently extended scale.

Generally speaking, the values of annuities on a single life from 3 to 8 per-cent, and on as many as six joint lives at any rate up to 4 per-cent may be deduced directly from the table. This will no doubt be found sufficient for most practical purposes; but by means of the property of the gamma-function before adverted to the application of the table, if necessary, may be extended to cases beyond the above-mentioned limits.

I believe myself entitled to claim the credit of having first pointed out the equivalence of annuities payable at various intervals, when the period of the next payment is undetermined; and the consequent error of the practice of substituting the equivalent annual payment for Half-yearly and Quarterly premiums in the Valuations of Assurance Companies by the customary method of classification according to age. Mr. Woolhouse has since shown. that in valuing (in the gross) a large number of independent annuities, payable at different intervals and at different periods of time, the employment of the continuous-annuity value is the strictly accurate mode of procedure (see his paper "On an Improved Theory of Annuities and Assurances" in vol. xv). Hence it would seem to follow that the tabulation of the continuous-annuity values, in lieu of the arbitrary ❝ annually-payable" annuities, would be more consistent both with scientific accuracy and practical convenience. When the very common question is put, how many years' purchase is a given annuity worth?—without specifying either the interval or the period of payment, the true answer, and the only true answer, is, the continuous-annuity value for the given age.

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In the continuous system the "force of discount" becomes the nominal rate of interest; and would therefore be quoted in integers or the aliquot parts of integers. Thus (to take the case in Example II) in lieu of 03922 ... (ad inf.) which is the force of discount corresponding to 4 per-cent, we should take d=04. The value of the reversion on a life aged x at the nominal rate of 4 percent (continuous) would therefore be 1-04 xa, the numerical value of which expression may readily be ascertained by inspection. from the table of annuity values. Many other advantages of a practical character would follow from the adoption of the continuousannuity basis (as suggested by me in a letter published in the Journal for July 1861) for the valuation of life contingencies.