Before I proceed to the more immediate object of this paper, viz., to the Consideration of some leading Problems in the Theory of Probabilities, I would premise a few observations with reference to the doctrine of Life Contingencies.

In my work on Notation, p. 13, I have asserted a fact which is of such essential importance in the elucidation of the doctrine, that it can scarcely be too often repeated, or too strongly impressed upon the mind of a student, viz,, that curtate expectations of life are the values of annuities which do not bear interest; and this is universally true, whether they be the expectations of single lives, of joint lives, of the longest of two or more lives, or of a life or lives after another or others,—in short, however remote or complicated the contingencies may be in which they are involved; and the values of annuities, or those series in which interest is involved, are invariably in the same form of expression as their corresponding curtate expectations, as thus employing the notation explained in the work already alluded to.