The investigation of the laws by which the general mortality of the human species appears to be governed is of equal importance to the statesman, the physician, the natural philosopher, and the mathematician; and as you have had occasion to pay particular attention to the subject, I trust that it will not be disagreeable to you to receive the results of an inquiry, into which I have entered, for the purpose of appreciating, if not of reconciling, the many discordant opinions that have been advanced respecting the comparative mortality of mankind, at different times and under different circumstances.

Take first the case in which. A has not a life interest, but one for n years, remainder to B. Let P be a sum of consols, 100–δ the present price of consols; consequently, P consols sells for

cash, which may be called M; A's and B's shares of it being M1 and M2.

B has a right to a sum of P consols n years hence: it matters not to him how this sum arises, provided it be ready for him in n years’ time. Suppose such a sum now appropriated to accumulate for n years as will produce P, B will be in the same position as if A enjoyed the sum intermediately; and if such sum be now handed over to B he will neither gain nor lose. Supposing M2 to be this sum, M2 invested to accumulate for n years = P consols.

Suppose that the portion of surplus allotted to a particular policy at the division of profits, is applied so as to relieve the assured from the annual payments after arriving at an age to he determined by the amount of that portion or cash bonus, the other conditions of the policy remaining the same; or again, suppose that the cash bonus is so applied as to convert the assurance into an endowment assurance, payable at an age to be determined:–it is proposed to investigate formulæ to apply to these two cases.

In the first case, the cash bonus must be equal to the present value at the time of division of profits of a deferred annuity, of which the annual payments are equal to the premium for the assurance, the number of years for which the annuity is deferred being unknown; and the problem, in fact, is to find the number of years.