The mortality statistics of the Scottish Assurance Offices, whose experience it is proposed to develop before combining it with that of the English contributing Offices, have, after passing through various committees, and receiving the benefit of their consideration, eventually passed into my hands for final adjustment and development. Having accordingly considered the subject thus confided to my professional care, I venture to lay before the Institute a brief statement of the views which have occurred to me in regard to the processes which it is proposed to adopt for the arrangement of these data and transforming them into that shape which it is necessary they should assume before becoming the subject of comparison or of calculation.

There is no country in which, from a combination of circumstances, a census, if accurately taken, is more likely to afford novel and interesting materials than the United States of America. The marvellous progress of the nation, which, in the last 70 years, has increased in numbers from less than 4 millions to nearly 31½ millions in 1860, is alone sufficient to suggest changes in population, in wealth, commerce, cultivation of land, means of transport and circulation, growth of towns, &c., which, if the facts were carefully examined, would throw light upon many obscure questions in social and political economy, and even by the disturbance of their ordinary laws modify and correct the theories which have been mooted.

We have now to apply the principles laid down in the previous sections to the construction of tables, first of rational, and secondly of irrational, functions. It is only formations of the class last mentioned that are of real importance. A little previous attention to the class first mentioned, however, is desirable, as affording an opportunity of elucidating, under elementary conditions, the arrangement of the work which is found most convenient in practice. The present section will, therefore, be devoted to the formation of successive values of rational functions.

In order to complete the consideration of this question as far as it relates to a single life, I will now give some numerical examples showing the magnitude of the corrections, or the additions to the value of annuities when payable half-yearly, as found from the formulæ of Mr. Baily and Mr. Woolhouse. Those formulæ will also be exhibited under some new aspects.