Although, as I have said, observations upon assured lives (in my opinion) afford, upon the whole, by far the best examples of the typical characteristics of the normal law of mortality, yet it is evident that all such observations are nevertheless affected to a considerable extent, first, by the selection exercised by the office in regard to the admission of lives, and, secondly, by the selection exercised by the policyholders in the option which they exclusively enjoy of terminating the contract of assurance whenever their convenience or caprice may induce them to avail themselves of the privilege. While the form of selection last mentioned is probably much the more important of the two in its effect upon the aggregate mortality, yet, for obvious reasons, its effect is of minor importance considered as a “disturbance” of the normal law which expresses the relation connecting rates of mortality at successive ages with each other. The number of persons withdrawing from time to time at particular ages must necessarily be, more or less, proportional to the number existing at those ages among the entire body, but there can be no connection whatever between the number of persons admitted from time to time and the corresponding existing number. Hence, although the effect of withdrawal upon the aggregate rate of mortality is considerable, its operation upon the mortality table is throughout pretty uniform, and consequently it does not distort the table to anything like the same extent as the effect of selection in regard to admissions. For instance, at the earlier ages, say under 30, the lives recently admitted necessarily constitute nearly the whole of the lives upon which the observations at these ages are based; but this is evidently not the case at the middle and higher ages, the proportion of lives newly admitted being at this part of the table comparatively small.

I have long had an idea that the ordinary formula for the value of an annuity,

might be transformed into some such form as the following:

where the coefficients would represent the condensed effect of the mortality, and the powers of i would make the calculation easy and rapidly convergent. I have lately succeeded in making the proposed transformation.

Probably in no branch of actuarial work have American experts more conspicuously shown those qualities of ingenuity and originality which are freely recognized as among the most prominent characteristics of their fellow-countrymen, than in that which may be summed up, generically, in the title, “Mortality Experience.” And it is remarkable how many instances of their enterprise in this respect have, within the last few years, been en évidence, whereby our available information on vital statistics has been enriched. But a few years ago, the valuable work entitled System and Tables of Life Assurance, prepared and compiled under the superintendence of Mr. Levi Meech, was published, and the equally interesting, though perhaps less important, Experience of the Connecticut Mutual Insurance Company followed shortly after, in 1883. Other publications of a similar character have been circulated since that time, including the Experience of the Provident Life and Trust Company of Philadelphia, reviewed by Mr. Sprague (J.I.A., xxvi, 316); and the list has received its latest addition in the instructive volume whose title appears at the head of this article.