take it up and develop it, cacli commencing where the previous one had left off. The result has been, as stated above, that great advances have lately been made in the theory. It may be truly said that the recent advances and improvements in the theory of life contingencies have rendered all the existing text-books anti quated ; and until a new one shall be produced, bringing the treat ment of the subject down to the present time, a complete know ledge of it can only be gained by a diligent study of the Journal of the Institute of Actuaries and Assurance Magazine.
As intimated above, our remarks on annuities involving more than one life will be very brief. The methods em ployed for the calculation of single life annuities are easily extended to the case of joint life annuities. The funda mental equation
a &eq vp(l + *a)
is true of annuities on two, three, or any number of joint lives, if we consider/) to denote the probability that they will all survive for one year ; and l a the value of an annuity on the joint continuance of lives which are severally one year older than those on which the required annuity de pends. Thus we have x, y, 2, being the ages of the nominees—
and a,, Jt &eq vp m p t p, (1 + l a xyi } .
The columnar method of calculating annuities admits also of being extended to annuities on joint lives. In the extensive tables contained in D. Jones s work,
Djcy " IJ-yV", y being the older of the two ages,
where n T) xy is used to denote T) x+n . y+n .
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An annuity on the lasfi survivor of two lives, x fo^ a x + a y - a I}> and y, An annuity on the lasO survivor of three lives, a^ji eq a x + a y + a,- a a , - a, x - a xy + a xy , x, y, and z, ) An annuity on the last two \ survivors of the three \a x j t eq a 1/ , + a 1JC + a xy -2a X!/ , lives, x, y, z, )
If we have the values of annuities on the last survivor of two lives tabulated, as is the case in the Institute of Actuaries Life Tables, we may find the value of an annuity on the last of three lives by means of the formula a= rt + ( i* rt r. where w is found by means of the relation = *-,; see Ass. May., xvii. 266, 379.
The methods of approximation given by Lubbock and Woolhouse also apply to the calculation of annuities on the joint existence of any number of lives; see the latter s explanation of his method, Ass. Mag., xi. 322, and for an illustration of its application to complicated cases, xvii. 267. They may also be applied to find the value of an annuity on the last survivor of any number of lives; see Ass. Mar/., xvi. 375.
The formula usually given for the value of a reversionary annuity on the life of x to commence on the death of y is a x - a xy . But this is not sufficiently correct, being de duced frbm suppositions that do not prevail in practice. It assumes the first yearly payment of the annuity to be made at the end of the year in which y dies, and the last at the end of the year before that in which x dies; whereas in practice the annuity runs from the death of y, the first yearly payment being made one year after such death, and a proportionate part being paid up to the date of ar s death. A more correct formula, as given by Sprague (Ass. Mag., xv. 126), is *. x (. If the annuity is payable half-yearly, 1 4. f 1 4- f\* the value will be approximately (a x - a xu ] - ; and if jj quarterly, (a z -a^) / 1 , -a , I n practice, it is often sufficient to deduct half a year s interest from the value found by the formula a x a xy , when the annuity is payable yearly, a quarter of a year s interest when it is payable half-yearly, and an eighth of a year s interest when quarterly.
In dealing with annuities in which three lives are in volved, we are met by the difficulty that no tables exist iu which the values of such annuities are given to the extent required in practice. Such tables as those computed for the Carlisle 3 per cent, table by Herschel Filipowski are of too limited extent to be of any practical utility; for the values being given only for certain ages differing by multiples of five years, a considerable amount of labour is required to deduce the values for other ages. "When, there fore, we desire to find the value of an annuity on the joint lives of say x, y, and 2, it is usual to take the two oldest of the lives, say x and y, and find the value of a xy , then to look in the table of single life annuities for the annuity which is nearest in value to this, a a suppose, and lastly, to find the value of a wi , and use it as an approximation to that of rt^,. De Morgan, in a paper written for tho Philosophical Magazine for November 1839, and reprinted in the Ass. Mag., x. 27, proved that the value of or,,, thus found would be strictly accurate, if the mortality followed the law known as Gompertz s; that is to say, if the number of persons living according to the mortality table at any age, x, could be represented by means of the formula dg q . Gompertz proved, in the Philosophical Transactions for 1825, that by giving suitable values to the constants, the above formula might be made to represent correctly the number living during a considerable portion of life, say from age 10 to 60; but in order to represent by the same formula the numbers living at higher ages, it is necessary to give fresh values to the constants; and the discontinuity thence resulting has always been a fatal obstacle to the practical use of the formula. It has, however, from its theoretical interest, attracted a great deal of attention from actuaries ; and numerous papers on the subject will be found in the Assurance Magazine. A claim to the inde pendent (if not prior) discovery of the formula has been put forward by Mr T. lv. Edmonds; but this claim, respecting which many communications will be found iu the Assurance Magazine, is generally repudiated by competent judges. De Morgan further showed (Ass. Mag., viii. 181) that if the above property holds good, or a xy , = a vl , then the mortality