All tabular numbers drawn directly from observations, especially those appertaining to statistics, are more or less affected by unavoidable imperfections arising from defective information, insufficiency in the number of observations recorded by the experience, and other known or unknown incidental causes. When a consecutive series of tabular numbers is taken and differenced up to a certain order, the existence of the imperfections or errors alluded to, is at once revealed by the differences, which exhibit a conspicuous disturbance of their law of progression. To eradicate the errors and effect a proper adjustment of the original numbers is, however, an operation of peculiar and somewhat novel difficulty, and a critical examination of the empirical expedients, usually resorted to for such purpose, will show that instead of meeting the indispensable requirements of the problem, they only evade its most essential conditions, and thereby impregnate the results with a new series of theoretical errors, which too readily evade detection in consequence of the systematic uniformity of their law.

Some years ago I read before the Institute a paper bearing on this subject (see Journal, vol. xi., p. 90) in which I examined at considerable length the method of valuation which has been on various occasions so strongly advocated by Mr. Tucker, and which is denominated by him the “reinsurance method.” This term, however, appeared to me unsuitable, and I termed the method the “hypothetical method” of valuation; because the sums assured and the premiums are valued by means of a hypothetical table of the values of reversions and annuities, deduced by an inverse process from the premiums actually charged by the office. In that paper I compared Mr. Tucker's method with the net-premium method of valuation, and gave my reasons for believing the latter to be not only greatly superior to the former, but the method that should be generally employed. Subsequent consideration has however satisfied me that the net-premium method of valuation is also open to very serious objections, and that it is not applicable in all cases; and I propose on the present occasion, first to point out some of these objections, and then to consider briefly what method of valuation should be followed when the net-premium method is inapplicable. The subject however is a very wide one; and the present paper must be considered rather as a contribution to the discussion of a question that is all-important to those who have the responsibility of advising as to the liabilities of Life Insurance Companies, than as claiming to be itself a complete essay on the subject.

The annexed Table contains the values of the function for all values of x differing by ·01, from 0 to 10·4.

To find the value of the function for any intermediate value of x, enter the table with the value of x nearest to the given value, whether greater or less, and take out the quantities Δ and Δ′ opposite the interval to which the given value belongs.