source whatever out of which to provide for the reduction in value of reversionary annuities. In the office I am at the head of, I debit no interest at all; and then when a reversion does fall in, there is a considerable amount of profit which is available for any purpose required, or which may be simply carried into account as balance of profit and loss. With regard to the two classes of borrowers mentioned by Mr. Bailey, it will be noticed that my paper is confined to the consideration of a reversionary life interest, properly so called. A tenant in tail does not properly come under the same class as a borrower who has only a reversionary life interest, because we know that by legal arrangements lawyers can give us a charge upon the fee simple, provided the tenant in tail should survive the tenant for life.

The PRESIDENT-What you call a "base fee."

Mr. SPRAGUE-Yes, the tenant in tail can at once, by executing a disentailing deed, obtain a "base fee," or an interest in the estate, which lasts so long as he or any of his descendants (or male descendants if the estate is settled in tail male) shall be livingsubject, of course, to the interest of the life tenant; and when the tenant for life dies, the tenant in tail, by executing another deed, can acquire the fee simple of the property. The practical means by which a borrower's "base fee" can be most effectually made a security for an advance is a matter of so much interest that I may not, perhaps, be occupying your time too long if I describe it. The borrower, having cut off the entail and acquired a base fee, executes a power of attorney, authorizing the lender and his solicitor, and perhaps half-adozen of his clerks, or any one of them, as soon as the tenant for life is dead, to execute the necessary further disentailing deed. Then that deed is prepared, and as soon as ever intelligence is received of the death of the life tenant, one of the persons named in the power attorney executes the deed, and the thing is done without any further consent on the part of the borrower being necessary. Several names must be inserted in the power of attorney to provide against the risk of the attorney dying before the life tenant. The risk of the borrower dying before the life tenant is met by effecting insurances on his life against that of the life tenant; and the very remote risk of his dying after the life tenant, but before the disentailing deed is executed (as, for instance, both being drowned by the upsetting of a boat, and the surviving the elder for a quarter of an hour), is met by making the risk under the contingent policies last for, say, three months after the death of the life tenant.

younger man

With reference to the formula—

1−(P+d)(1+αxy)

of

Mr. Bailey remarks that it is Mr. Jellicoe's formula, and only applies when an annuity is actually bought; but if he looks at it a little closer he will find it is exactly the same formula as is really used in the process he describes, the only difference being in the rate of interest. In Mr. Jellicoe's method, d is taken at 5 per-cent, and the annuity on the joint lives is taken at 3 per-cent, say; but if you take d at 6 per-cent, and the annuity on the joint lives also at 6 percent, that is exactly the same thing as the ordinary formula for the value of a reversion at 6 per cent. For

Axy=1―d (1+axy), and A'xy=P'xy (1+αxy).

It is curious to note how the same identical formula will bring out different results, the difference being in the rate of interest. If, for instance, in the above formula P were taken as the net premium, the formula would be the absolute bare value of the reversion at 6 percent. The case which Mr. Bailey has mentioned, of the office undertaking to grant a whole-life assurance on the death of the tenant for life, whatever may then be the state of the borrower's health, is one which he will see I have fully considered in the paper. I now pass on to consider what Mr. Macfadyen has said. He does not quite see, I think, the use of the optional increasing assurance I have proposed. If you have an ordinary increasing assurance, no doubt if the borrower wants the insurance, the cost of it to him beyond the net premium is less than that of the optional increasing assurance; but the fact is, in many cases he does not want the increase of the insurance at all; and it is not, as Mr. Macfadyen supposes, like having a picture given him for part of his loan, for if the picture is a good one it will rather increase in value in time, but if a man has a large increasing assurance, he pays his premiums for the current risk, and has nothing to show for them afterwards. So that when a man is forced to effect such an assurance, which he does not want, it is no answer that it is an increasing assurance and he has had the benefit of it. He is exactly in the position of a man who is obliged by a money-lender to take a large quantity of wine that he does not want, which he cannot sell, which will turn sour if not drunk, and which he is therefore constrained to give away to his friends. By effecting this optional increasing assurance on the contrary, he knows the worst of it, and he has to pay a certain consideration, 5s. per-cent per annum. company gets that consideration; the man pays it and does not grudge the money, because he gets the insurance he wants, and is saved the larger outlay for the increasing insurance which neither he nor the office wishes for, and which is of no value to anybody except his heirs if he should chance to die before the loan is paid off. So that the optional increasing insurance appears to me to give exactly what is wanted, and at the lowest possible price. The question mooted by Mr. Baden as to the character of the borrower is no doubt one of great importance, and has to do not only with reversionary life interests, but with life interests in possession. It is necessary in dealing with both of them to take care that you leave a sufficiently large margin to make it worth the while of the borrower to come forward from time to time and claim the surplus. If proper care is taken to have a margin, you may be sure, when the tenant for life dies, the reversioner will come forward and claim his interest. Mr. Tucker has told us that he never heard of the case of an annuity being actually bought. I do not know how it may be with regard to reversionary annuities, but I am informed there is one large old insurance company in the city that always buys an annuity when it buys a reversion, and thus entirely gets rid of the difficulty as to the interest account.

Mr. BAILEY-Buys it from itself?

The

Mr. SPRAGUE-No, it does not grant annuities; but actually purchases the annuity elsewhere.

On the Arithmometer of M. Thomas (de Colmar), and its application to the Construction of Life Contingency Tables. By PETER GRAY, F.R.A.S., F.R.M.S., Honorary Member of the Institute of Actuaries.

THE Arithmometer of M. Thomas (de Colmar) has been already brought under the notice of the readers of this Journal by General Hannyngton, in a remarkably lucid and suggestive paper, which will be found at p. 244, vol. xvi. General Hannyngton, in his paper, explains the manner of working the machine, and gives examples of some of its applications to the construction of actuarial tables, with hints as to others. These afford an idea of the very striking adaptation of the machine to the formation of such tables; and they cannot fail to have excited the interest of many of the readers.

The present paper is intended to be supplemental to that of General Hannyngton, and in it the adaptation in question will be further shown. An attempt will also be made to systematize the månner of its application, and detailed examples in illustration will be given. There will be no need to say anything here as to the actual working of the machine, since this has been so well explained by General Hannyngton. There are, nevertheless, certain points in the manipulation to which it may be well to advert in the outset.

It is usual to describe the Arithmometer as a machine which enables a person, however unskilled himself, to perform the operations of multiplication and division with facility, rapidity, and unfailing accuracy. This, as a description, is correct as far as it goes; but as an enumeration of the properties of the machine, it is inadequate and defective. It entirely omits that property which forms its special adaptation to our purpose, and in default of which its utility would be comparatively limited. Besides the facilitation of the operations named, the machine will also, in forming the product of two given numbers, either add that product to, or subtract it from, another given number, according to the pleasure of the operator. Abundant illustration of the application of this property will be found in the present paper.

There are three forms, then, to the numerical evaluation of which the Arithmometer is directly applicable. These may be symbolized as follows:

Q

QR,

and P+QR;

R'

P, Q and R denoting any given numbers.

*An illustration of it is given by General Hannyngton, in the formation

of Dx.y.

Familiar instances of these forms, in the present connexion,

The manner of applying the machine to the evaluation of expressions of the first and second forms; in other words, the manner of performing the operations of multiplication and division, has been sufficiently explained in General Hannyngton's paper. Of its application to the third form some further elucidation is necessary. The special adaptation of this form to the construction of tables may be shown as follows:

If u be any function of a, we always have,

Ux+1=Ux+Aux;

and this is of the form P+QR if Aux, the difference of ux, can be exhibited as the product of two given numbers. In such cases, then, (and it will soon appear that they are by no means rare), we have, in forming a series of successive values, the benefit of a continuous process, with its attendant advantages. The result of each operation becomes the P of the next, and we pass from a preceding to a succeeding value by adding to (or subtracting from) the former the product of two known numbers. And this, as we have seen, is a function to which the Arithmometer most readily lends itself.

In the applications of the machine there are thus, it appears, three numbers to be dealt with; and provision is accordingly made on it of three spaces for their separate exhibition. Two of these spaces are on the slide, and the third on the face. I propose to designate them by S1, S2, and F, respectively; and the numbers occupying the several spaces will be denoted by the same symbols, respectively distinguished, however, when so used, by being enclosed in parentheses, thus :-(S1), (S2), (F).

In employing the formula P+QR, the number P is placed on S1, and Q (the in factor), upon F; multiplication being then made by R (the out factor), this number appears upon S2, and the result, which is the required value of P+QR, replaces P upon S1.

A point in regard to which difficulty is felt, in commencing the use of the Arithmometer, is the setting of the numbers P and Q upon the machine, so that the product QR shall, when formed, fall

in its proper place with respect to P. The rule for this purpose (which has not hitherto been given) is very simple. It is:

Draw out the slide as many holes as there are decimal places in R, the out factor; and place P and Q upon S, and F respectively, in such a manner as that like denominations shall stand under (and over) like. The slide is then to be pushed home; and, the multiplication being performed, the correct result will appear on S1. If P=0, that is, if it is only the product QR that is required, no preliminary drawing out of the slide is necessary.

So far as the results of the formulæ QR, P+QR are concerned, it is obviously indifferent which of the two, Q, R, we employ as the in factor; in each case of practical application, however, there are usually circumstances sufficient to determine our choice in favour of one or the other. I shall in general, when necessary for distinction, let Q denote the in factor-the factor to be set upon F; and R, of course, to denote the out factor-the factor to be employed as a multiplier.

The points that have been now more specially adverted to, and others that may arise, will find ample illustration in the examples appended to the problems to which I now proceed.

The examples will be taken from The Institute of Actuaries' Life Tables, H mortality, and three per-cent interest.

PROBLEM I.-Given a table of annuities; to form the corresponding table of assurances.

We have,

A2=1− (1 − v) (1+ax) ·

AA-(1-) Aαx;

(1)

(2)

whence,

Ax+1=Ax+▲Ax=Ax−(1—v)▲ax ·

By these formula the required table will be constructed, (1) coming into use for the formation of the initial term, and (2) for the continuous formation of the subsequent terms.

Example 1.—The given table of annuities is that on single lives, p. 14 (Institute Tables); and it is required to reproduce the table of assurances in the adjoining column.

Here we have, by (1),

We here take 1-v for Q, because it remains constant during the process; and we use six significant figures because, there being