^{page 249 note *} An illustration of it is given by General Hannyngton, in the formation Of D_{X y}.

^{page 252 note *} The method here made use of for the correction of the last figure in the terms of series formed by addition (or subtraction) was, I believe, first suggested and exemplified by me, in a work published in 1849. Among the instances of its advantageous employment that have come to my knowledge, the most striking is one that occurred to Dr. Farr, in the construction of his tables of log v^x (Tables of Life-Times, pp. 6–11). These tables were formed by the aid of Seheutz's machine; a specialty of which is, that it records its results in the form of a mould to be employed as the matrix for a stereotype cast. It is consequently necessary, in the use of this machine, that each result, as it arises, should be correct in the last figure to be recorded. Dr. Farr informs us, in his Introduction (p. cxliii), that the method he employed for this purpose was that now under consideration. The required correction could probably, in this case, have been given in no other way.

^{page 253 note *} The comparison here suggested will reveal the existence of a few slight discrepancies in the last place between the two sets of values compared. Both nevertheless are correct deductions from the data employed in their formation. The discrepancies, such as they are, originate in the relation which subsists between the values of an annuity and an assurance on the same status. It can easily be shown that, at three per-cent, four-decimal annuities are barely sufficient for the accurate determination of six-decimal assurances.

^{page 254 note *} The facility with which errors of this kind, when noticed at tie time, admit of correction, forms one of the great merits of the Arithmometer. The erroneous figure being brought into the working position, it is set right by the requisite number of turns; remembering that, preparatory thereto, if it is diminution that is required, the regulator must be reversed.

^{page 255 note *} I find that the symbol for the value of a policy, _n V _x , given in the recognized notation, is unsuited when, as in the present investigation, it is necessary to treat the value as a function of the ages at entry and at valuation. I have therefore been obliged to devise a new symbol; and the one in the test, while answering my present purpose sufficiently well, is so distinctive that there is no risk of its being confused with any other symbol.

^{page 257 note *} The reciprocals for the entire annuity column had better be formed at the outset, either by the machine or by Oakes's Table.