Life Assurance is an institution possessing such an important influence in the welfare and well-being of Society that it is most essential the principles upon which it is based and conducted should be such as to ensure its lasting stability and prosperity.

After a Life Assurance Office has been once established upon a firm basis, there is no part of its management so important as that relating to the periodical valuation of the liabilities under its policies. It is my object in this essay to inquire into the various methods adopted for that purpose in the present day; and to consider the various results they produce when different tables of mortality and rates of interest are employed in the calculations; and also to ascertain the effects which the employment of different data may have upon the present and future condition of an Office.

Irrational Functions, strictly speaking, are functions whose numerical values cannot be expressed by finite fractions. In the present connexion the term irrational has a wider signification. It is used to denote any function whose numerical value cannot, in general, be expressed exactly within the limits as to decimal places, to which we restrict ourselves in the table under formation. As here employed therefore the term designates not only transcendental functions, (as exponential, logarithmic, circular, &c.,) and algebraical irrational functions, (which are such as contain fractional powers of the variable in either numerator or denominator, or both,) but also algebraical fractional functions, which are such as contain integer powers of the variable in both numerator and denominator, or in the latter only. And it is to the formation of tables of the values of functions such as these that our attention is now to be directed.

On the 1st February, 1868, this Company, “now the largest moneyed Corporation “in America, celebrated its “silver wedding,” as it was termed by the Governor of the State of New York on the occasion of the 25th annual meeting. The Report then presented and the general Prospectus of the Company have been forwarded to us by the “President” of the Company, who appears to be virtually what in this country we should call “Managing Director.” These documents are so instructive and interesting in themselves, and illustrate so remarkably the application to our pursuits of that originality of the American character, which is fast becoming as proverbial as its energy, that we propose to notice their contents in some detail.