In the correspondence that has recently taken place respecting the origin of Gompertz's formula expressive of the law of mortality, it has been stated, much to my surprise, that the investigation first given by that gentleman in the Philosophical Transactions of the Royal Society for the year 1825 was defective and inaccurate. I do not here propose to reopen the controversy on the subject of the originality of discovery, or to enter upon a renewed discussion of certain other allegations that have been so efficiently disposed of by Professor De Morgan and Mr. Sprague; but, after the statement to which I have more particularly alluded, it is only right that I should avail myself of the present opportunity to communicate the circumstance that Mr. Gompertz kindly presented me with a copy of his valuable Memoir in January, 1839, in which the various misprints had, in the most legible manner, been previously corrected by himself, and that the processes so corrected are perfectly accurate in every respect. In order that any one may be enabled to judge of this for himself, I shall here give a faithful transcription of that portion of the Memoir which contains the mathematical investigation of the formula, as it will occupy but little space.

The system pursued by each Office in the distribution of surplus may be divided into three parts, namely—

1. The class of members amongst which surplus is distributed;

2. The actual distribution of it amongst that class; and

3. The interval which elapses between successive distributions.

It is impossible to obtain a comprehensive view of the subject without considering separately these three distinct parts; and, since it forms no part of the present purpose to estimate the relative advantages of different Offices, no attempt will be made to follow the numerous combinations or systems of distribution produced by differences in practice under each of the above heads.

Of the many valuable contributions by which the transactions of this Institute have been enriched by the labours of Mr. Samuel Brown, not the least interesting is the paper on the investments of Assurance Companies, read before the Institute on the 25th January, 1858. I feel much diffidence in again reverting to this question; but as Mr. Brown, at the outset of his paper, admits that the subject is too extensive to be exhausted by a single discussion, it has occurred to me that an attempt to arrive at some general principles on which the funds of Life Assurance Societies should be invested, and to consider how such principles can best be reduced to practice, might, perhaps, be the means of exciting some useful discussion.

To find the value of a life annuity on (x), of which the first payment is to be £1; the second, £2; the third, £3; and so on, increasing annually, by the amount of the first payment, to the end of life.