The subject of Graduation has recently been brought somewhat prominently before the profession in connection with the exposition of various methods that have been proposed. While the bases of these individual methods have been set forth in a way leaving nothing to be desired, the precise relation of each to the entire problem of the adjustment of probabilities derived from statistics, does not appear to have been so clearly marked out; and therefore, in the following observations, it is proposed to keep prominently in view the conditions under which one method is to be preferred to another.

An intelligent outsider, desirous of becoming acquainted with the actuarial theory of graduation, might, if he overlooked the necessarily limited application of particular systems, derive from his researches an impression of disagreement among the various authorities, which would have a rather bewildering effect.

There are several cases in which the factors of a number may be of use in ordinary mathematical processes; as for example, in the calculation of logarithms. This, although not the raison d’être of such a table, is so important an application as to deserve special notice, for by its means the number of numbers whose logarithms are known is greatly extended. For example, Abraham Sharp’s table contains 61-decimal Briggian logarithms of primes to 1,100, so that the logarithms of all numbers whose greatest prime factor does not exceed this number, may be obtained by simple addition. Similarly, Wolfram’s table gives 48-decimal hyperbolic logarithms of primes up to 10,009. Thus a factor table forms a very valuable complement to many-place logarithmic tables, the range of which must necessarily be comparatively limited ; and even in the case of large prime numbers, or numbers exceeding the limits of the factor table, it affords a very convenient method of calculating logarithms.

In my paper On the Construction ov a Combined Marryge and Mortality Tabl (J.I.A. xxi, 423), I shoed how the statistics suplyd by the Scottish Registrar-General may be employd to calculate the marryge rate among bachelors. I wil now use the same statistics to calculate the marryge rate among widos and spinsters. As I stated in the paper abov referd to, the Scottish marryge statistics apear to be furnisht with greater exactnes than the English, and the conclusions drawn from them as to the marryge rate, seem therefore much more trustworthy than the corresponding conclusions drawn from the English statistics.

In a book entitled Principles of the Algebra of Logic, which deals with the Sciences of Necessity and Probability as one whole, I have considered as an example the question in probabilities discussed in vol. xxi (p. 204) of this Journal, namely—“ A says “ that B says that a certain event took place : required the pro- “ bability that the event did take place, p1 and p2 being A’s and “ B’s respective probabilities of speaking the truth.”

When it is desired to disentail a landed estate, it is necessary for the heir in possession, after obtaining the consent of the first substitute heir, to pay to the second and third heirs the estimated value of their expectancy or interest in the estate. In the calculations that have to be made for the purpose of ascertaining this value, the actuary has often to take into account not only the probabilities of life, but the probabilities of marriage and of leaving issue.

Suggestions have been made at various times that the Directors of life insurance companies should encourage the practice of life insurance among the members of their own staff; and although we are aware that something of the kind has been done in a few instances, we do not know that this is at all general.