Gompertz's Function for expressing the probability of a person surviving anygiven period, x, is . For thepurpose of adapting this expression to the entire period of adult life Ihave added the factor ∈−ax. The two expressionsbecome identical in form when the factor∈−δx representing the effect of discount is introduced, and both may therefore becorrectly designated by the name of “Gompertz's Function.”

In the year 1865, the late Rev. John Hodgson published a pamphlet entitled “Observations in reference to the Duration of Life amongst the Clergy of England and Wales;” to which is added, in the form ofa supplement, a letter from Mr. Samuel Brown, containing various tables,having the observations of Mr. Hodgson for their basis, applicable toquestions connected with Life Assurance, and the Valuation of NextPresentations to Livings and Advowsons.

In recent years, the writer who has done the most good service in bringingscientific methods to bear on the collection and comparison of statistics,is M. Quetelet, the Director of the Royal Observatory in Bruxelles,President of the Central Commission of Statistics, and the PerpetualSecretary of the Academy of Sciences. Nearly all the learned Academies ofEurope claim the credit of enrolling him as a member. He was amongst thesmall number of illustrious men, including Professor Babbage, Whewell,Malthus, Drinkwater, Jones, Col. Sykes, &c., who at the third meeting ofthe British Association for the advancement of Science, at Cambridge in1833, succeeded in establishing the special Section F on Economic Scienceand Statistics, and which afterwards led to the formation of the StatisticalSociety of London and various provincial Societies. He was also theoriginator of the International Statistical Congresses, which have been souseful in improving the methods of collection and publication of facts inall branches of statistical enquiry by every Government in Europe.

The following results compiled from my paper “On the Progress to Accuracy ofLogarithmic Tables,” read before the Royal Astronomical Society, on 13thDecember 1872, will probably be found interesting to the readers of the Journal of the Institute of Actuaries, more especiallyin connection with the recent correspondence in reference to Mr. Sang'slogarithmic tables.

Besides the probable error above considered, there is another quantity ofsome importance in practice, which may be called the Risk of a departurefrom the most probable number of survivers. It is obtained by addingtogether the products of all the possible positive deviations in the numberof the survivers from its most probable value, each multiplied into theprobability of its occurrence; or by adding together the products of all thepossible negative deviations in the number of the survivers from its mostprobable value, each multiplied into the probability of its occurrence; inthis case, however, taken without sign. For if we consider the deviationsfrom the most probable number of survivers, taken without regard to sign, asrepresented by proportional sums of money, then the quantity just definedevidently expresses either the mathematical expectation of gain of a personwho is to receive those sums, or the mathematical expectation of loss of aperson who is to pay them. The latter of these two ideas more generallyoccurs in practice.