The two columns following that containing the ages, represent two cardboard slips, on which are written, in reverse order, the terms of and l, respectively. Their position with respect to each other and to the ages in the margin, is regulated in accordance with the formula of construction. Here we have, for example, opposite x=96, 97 and l97; opposite x=95, v96 and 196, and so on. The process now consists in the multiplying together of the corresponding numbers on the cards, in succession and the setting down of the results, as they arise, on the same line. These, as stated, are the terms of N corresponding to the values of x in the margin. It will be understood that in this process, in accordance with the formula of construction, the results are not removed from the machine, (S2) alone requiring to be effaced after each multiplication.

I say nothing here as to the method of procuring verification of the foregoing process. I defer this, with further remarks on the construction, till the next, following problem shall have been discussed.

Next, to form Dr. We have, as above,

x

Hence the Column D will be formed by differencing, in the usual way, the Column Ne, when written as above in reverse order. And the differences, when written as in the specimen, each opposite the subtrahend by the employment of which it is deduced, will be in their proper relation to the ages in the margin. And we have now, consequently, in line with x, ve+1, la+1, Nx, and Dr.

To form D independently of Nr, the same process as that for the formation of the latter column would have to be gone through, with the addition that besides (S2), (S,) also would have to be effaced after each multiplication. The process also would be discontinuous; and in consequence, unless the work were performed in duplicate, no verification could be procured till both Nr and Mr should be in course of formation. If to this it be added that the deduction of Da from Na is at least as easy as that of Nr from Da, sufficient will have been said to make it manifest that, in the construction of a Commutation Table by the aid of the Arithmometer, the proper course, in regard to the annuity columns, is to commence the formation with N. In the case of the assurance columns also, corresponding advantages attend the commencement of the formation with Mr.

PROBLEM IV. Given v and de; to construct Columns Ma

and Cz.

Since,

Hence,

Mx=Cx+Cx+1+.....

AMx=-Cx=-vx+1dx.

Mx+1=Mx+AMx-Me-v+dx.

Column Mx, like Column Na, and for a like reason, must be constructed in reverse order. Accordingly, transposing, we have, Mx Mx+1+x+1dx,

=

a formula for the construction in such entire accordance with that of the last problem, that little more is necessary than to direct attention to the specimen here given.

The cards here are v and de, and their disposition, to accord with the working formula, is somewhat different from that of the cards in the last problem. Thus, in line with age x we have here +1 and de. When so arranged, the process for the formation of M is identical with that of last problem for the formation of Nx.

Hence the column will be formed by differencing, as they stand, the terms of Ma; observing that here the differences as they arise are to be written, not as in the last problem, opposite the subtrahends, but opposite the minuends. When so written we have, in line with a, v+1, de, Mr, and Ca.

It has not been usual hitherto to tabulate Cr. But this column can be turned to such good account, for the formation of various subsidiary tables by the aid of the Arithmometer, that it will probably come to be tabulated, in a special category, along with others similarly adapted, such as Aax, Dx-1, Nx-1, &c.

When we multiply together two factors, of which one is a nonterminating number, a portion of the result has to be rejected, as not properly belonging to the product; since it would be altered if the non-terminating number were further extended. And therefore it is that in the process of contracted multiplication labour is saved by so arranging the work that only the correct portion of the product is formed. The Arithmometer attains the same end in another way: in the use of it we form the entire product of the numbers submitted to it, and we neglect the useless figures in recording the results. It is therefore desirable to be furnished with a guide as to the extent to which this process of curtailment ought to be carried.

In the case supposed the multiplication of a terminating and a non-terminating factor-we know that we can depend upon about as many figures in the product as there are of significant figures in the non-terminating factor; and hence in the several terms of Da and C, we shall have as many places true as there are in the terms of which enter them, respectively. Also, in the several terms of N. and Mr, which are respectively summations of series of terms in Dr and Cx, we should expect to find one or two more figures correct than in the corresponding terms of D and C.

x

We may say, then, that in the formation of N and M, it is unnecessary to record any term to more than two places beyond the number of significant figures in the power of v in immediate use.

We shall find these conclusions verified in the case of the columns before us. I have used in the formation the column v (3 per-cent) as contained in Jones, vol. i, pp. 79 and 82, having first verified it by the aid of the Arithmometer. It contains seven significant figures at the outset, which further on are increased to eight; and I have used it to its full extent for experiment, although generally one or two places fewer will be considered sufficient.

x

Corresponding values in N and M are connected by the equation,

The commas indicate the extent to which the values thus formed agree with those formed by the use of the machine. The agreement in the four examples extends to seven, eight, and nine places, respectively; but when we attend to the manner in which the formula lends itself to these deductions, we shall not fail to see that in each case the correctness of the Ns concerned is established generally to two places more. For example:

M19

18

Ten places in N1s and N19 we perceive here come into use for the determination of the eight correct places in M19. We conclude, therefore, that at this part of the table Ne may be depended on to ten places of figures, and no more than this number need have been recorded.

Also, if the work has been correctly performed, there appears why Me should not be accepted as true to the same extent as N. As to the number of places to be finally tabulated

no reason

for use, the computer will, of course, follow the dictates of his own judgment.

By the use of the foregoing formula, N and Ma may be verified as the work proceeds. Both columns being brought down to the same point by the formation of, say, twenty terms in each, comparison can be made at this point as above shown. If found satisfactory, the formation may be continued till another point of comparison is reached; and so on till the columns are completed.

It is hardly necessary to point out that the contrivance formerly suggested for correcting the last figure retained in the several terms, cannot in these constructions, be advantageously applied; the reason being, that here the last figures vary in their distances from the decimal point.

The pegs also are employed here in a manner different from that in which they came into use in Problems I and II. They are now employed to facilitate the recording, by separating the results on S, into convenient periods.

(To be continued.)

Note on a Method of finding the Value of an Annuity on the Last Surviver of Three Lives.

MR. William Godward, of the Law Life Office, has communicated to us a method by which the values of annuities on the last surviver of two lives contained in the Institute Life Tables may be very conveniently applied to calculate the values of annuities on the last surviver of three lives.

Let x, y, z be the three lives, of which a is the youngest, and let w be the single age equivalent to the joint lives y and z (so that away), then the value of the annuity on the last surviver of x, y, z may be found by the formula

axyz=αxy+axz-axw•

The truth of this formula, which Mr. Godward informs us is given by Simpson in the Supplement to his Annuities, p. 58, is easily demonstrated.