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Published online by Cambridge University Press: 18 August 2016
We have now to give some examples of compound benefits, which are those consisting of two or more simple benefits; but the combinations which may be formed of these being obviously very numerous, it would be beside our present purpose to attempt giving a complete list of them. Our object will be, in selecting a few of them for illustration, to indicate the method of dealing with the more complicated cases, and also to prepare the way for the most general application of the Commutation Tables, which application will form the subject of the concluding portion of this paper. A very complete list of the formulae for the more elementary of these benefits is contained in Professor De Morgan's first paper on the subject; and as it is hoped that little difficulty will be experienced with these, after the illustrations to which our space limits us, we shall not scruple, as we have occasion, in the solution of any of the problems with which the present paper will be occupied, to refer to any of the learned gentleman's formulæ which we may not have deduced for ourselves. Our references will be made in the following manner, which is rendered necessary in consequence of the formulae not forming one consecutive series. Formula 10, on page 16, for example, will be denoted thus, [16,10]; formula 72, on page 18, thus, [18, 72]; and so on.
page 220 note * It may be of use here to point out a few typographical errors in Professor De Morgan's papers, which might otherwise embarrass the student:—
Also, the terminating braces are omitted in the expressions [18,11] and [19,152] of the first paper.
page 227 note * Mr. Davies' work having been long out of print, we have not been able to procure even a sight of it. But we have no reason to believe that it contains anything akin to the contrivance we have referred to. With the other works named on pp. 84, 85, as treating on the use of the Commutation Tables, we possess a competent acquaintance; and, certainly, in none of them is the least trace of this contrivance to be found.
page 231 note * The Commutation Table, as it stands, does not enable us, conveniently, to find the amount of annual premium equivalent to a benefit which consists partly of a return of all the premiums paid, with simple interest upon them from the date of payment, the incremental portion being in this case of the form m, 3m, 6m, 10m, &c The addition of another column, formed from R as R is formed from M, would, however, afford the means of doing so. W e might, obviously, add as many columns as we pleased in this way. Their properties would be such that, calling the column formed from R the first, the division of any number in the nth column, by the corresponding number in D, would give the present value of an assurance whose payments should be the series of figurate numbers of the nth order; and the remark may be extended, mutatis mutandis, to the annuity columns. But such properties being more curious than useful, we do not insist upon them. (See De Morgan, I., p. 23.)
page 234 note * This is by no means necessarily the case. The actual premium must always be used.