While life insurance premiums, irrespective of the additions for the cost of management, are so calculated, that assuming a given mortality table and a given rate of interest, the receipts of the Insurance Office improved at compound interest will exactly suffice to pay the claims that occur according to the table of mortality; yet the Office suffers a loss in those cases where death occurs soon, and on the contrary makes a profit in the case of the persons who live long. In the grant of life annuities, the contrary is the case. It can be calculated exactly how long an insured person must live in order that the Office may make neither profit nor loss on his insurance; and in the same way it can be calculated what gain or loss the Office experiences when the life assured dies after a given number of years, n. If all these gains and losses are added together, the losses being considered as negative gains, the sum must be = 0. This, however, is far from being the case in practice. The datums of the mortality table, altho' they may be deduced from observations extending over a long series of years, are to be considered as mean values, which are strictly applicable only when there is a very great number of deaths; and we ought to expect that with a small number of insurances, and still more in single instances, considerable deviations from the mean will take place.