The first line is occupied by log v97, and the fourth, seventh, tenth, &c., by colog v; the second line is occupied by log 197, and the fifth, eighth, eleventh, &c., by the successive terms of colog❜ in reverse order. Before proceeding to the additions it is necessary to introduce certain corrections. The true value of colog v is 012837,2247 ...., and the neglect of the figures beyond the sixth would give rise to constantly increasing error, which, at the close in log D10, would amount to no less than 20. To counteract this a periodical correction is made, and the points where it is to be introduced are easily determined by reference to the series log v2, a portion of which, in reverse order to correspond with the order in which the table is being formed, is given in

the margin. Commencing with log 97 (which is the term that enters into log D97) and attending to the last figures only, we see that the differences are 7, 8, 7, 7, 7, 8, 7, 7, 7, 7, 8, .... Let the terms on which the 8's fall be marked with +, meaning that those terms have been increased by unity. The marked terms here are 95, 91, 86, &c., and the terms in log D, corresponding to those ages will thereupon have to be similarly increased, since log D=log + log v*. The+marks are therefore placed against the proper ages in the working column, and it is then seen that what is requisite is to alter the value of colog v next preceding each marked line from 012837 to 012838. This correction being made† the addition is proceeded with, and the successive sums are the terms of the series log Da, true to the nearest unit in the last place.

It is obvious that the frequent writing of colog v may easily be avoided. It should be written at the bottom of a card; and this being held over the sum last formed, the logarithm can be included in the next summation. This method will be exemplified in the formation of log Cr.

The numbers corresponding to log D, were taken out by one computer and carefully examined by another. Further verification was obtained when Columns N and M were formed, as will be shewn hereafter.

The formation of a portion of Column N is shown; but it is unnecessary to exemplify that of Column S.

The method employed for the formation of Column C will now be described.

We have

Hence,

log C2=log v2+1+log dz· A log C,=log v+▲ log dr.

* A six-figure table of log, which shall be true in the last place, cannot be formed from a seven-figure table, since we are left in doubt as to whether or not a terminal 5 in the seventh place indicates an increase in the sixth. And in especial it cannot be formed from Mr. David Jones's values of this function, these being very incorrect in the last place, particularly the 4 per-cent. column. An excellent set of tables of log v*, extending to ten places, and therefore suited for every purpose, is given in Mr. William Thomas Thomson's Actuarial Tables, to which work, Mr. Thomson informs his readers, it was contributed by Mr. Filipowski.

Consequently, for the formation of log C, in direct order, we have

log Cr+1=log C2+ A log C2

=log C2+log v + ▲ log d2 ;

and for the formation in reverse order, which is the more convenient, we have

log C1=log C++colog v + A colog d2 ;

in which ▲ colog d, is the function occupying the column marked -A in the Elementary Table.

The following is an example of the formation:

The formation here is effected in the manner just referred to. The only addends written down are the terms of the series A cologd, (in reverse order), and colog v is included in each summation (after the first) by means of a card.

In consequence of the constant addend, colog v, not being set down, the corrections rendered necessary by its curtailment to six places must in this formation be applied to the terms of A colog dr. Their places are determined at once by reference to the formation of log D. In the present formation the marked terms are those corresponding to the ages one year younger than those to which they correspond in the lastmentioned formation. The powers of v which there enter into D, enter here into C-1. Hence, applying the marks, and increasing the addend immediately preceding each by a unit in the last place, continuous addition gives the terms of log C, true to the nearest figure.

The numbers being taken out and examined, the formation of Column M is proceeded with, and a specimen of the operation is exhibited.

We can now procure verification of all the preceding work.

Since

we have

M=UN2--N2,

M85 VN84-N85;

=

and the formula is applied as follows :—

103)1643354(1595490 VN81.

613 1203816

This process is easier than the logarithmic one, when, as here, division by I+i is used instead of multiplication by v. It also, usually, gives us one or two figures more in the result than can be obtained by the use of logarithms.

A similar relation holds between the Columns S and R. Thus,

And by the application of these formulæ the whole of the Commutation Tables in the present volume have been verified,

4. Logarithms and Cologarithms of the Principal Columns of the Commutation Table.

This table, which contains also the differences of the logarithms enumerated, occupies the four pages next following each of the Commutation Tables. The foundations of the table are the columns containing log D2, log N, and log My; and the manner in which the remaining columns are derived from these is sufficiently obvious to preclude the necessity for description. The columns were severally verified in the same way as the corresponding columns of the Elementary Table.

5. Results derived from the Commutation Table.

The results here tabulated are a,, A, and w, the values of annuities, and the single and annual premiums for assurance of a unit.

They were formed as follows:

A log

x

=log A,+▲ log M2+colog up.

= log M2+colog N2-1.

A log M2+ A colog N2-1

log @+1=log2+ A log ☎,

=log + A log M,+ A colog N1-1.

I