It is not intended in this paper to do more than go briefly again over part of the ground covered by Mr. Manly, in his admirable essay, printed in the 14th volume of the Journal of this Institute, on the Values of Policies as found by means of various Tables and Methods of Valuation.

It is now some seven years since Mr. Makeham pointed out that if our mortality tables were graduated by his formula, , “all summations connected with questions of population, the expectation of life, as well as the values of annuities, would be reduced” to the form . Since that time Mr. Makeham's formula, though not representing the law of mortality so closely as some others which have been or can be presented, has nevertheless, on account of its comparative simplicity and its great practical advantages, made manifest progress in the favor of the profession. Messrs. Woolhouse and Brown, among others, have constructed life tables in accordance with it, and it is understood that many actuaries favor its general adoption.

If the death take place in the (t+1)th interval in the year of death, then the present value of the t instalments of each paid in that year previously to the death, is .

The Contemplation of the Mortality of Mankind, has besides the Moral, its Physical and Political Uses, both which have been some years since most judiciously considered by the curious Sir William Petty, in his Natural and Political Observations on the Bills of Mortality of London, owned by Captain John Graunt. And since in a like Treatise on the Bills of Mortality of Dublin. But the Deduction from those Bills of Mortality seemed even to their Authors to be defective: First, In that the Number of the People was wanting. Secondly, That the Ages of the People dying was not to be had. And Lastly, That both London and Dublin by reason of the great and casual Accession of Strangers who die therein, (as appeared in both, by the great Excess of the Funerals above the Births) rendred them incapable of being Standards for this purpose; which requires, if it were possible, that the People we treat of should not at all be changed, but die where they were born, without any Adventitious Increase from Abroad, or Decay by Migration elsewhere.