The value ov the annuity calculated in the ordinary way is

16.4744.

Woolhouse's formula (6), being similarly delt with, becoms

(see Jurnl, xi, 320)

Making n=7, we hav the value ov the annuity

=16·6437-4 x '04241

=16.6437-1696

=16.4741.

Making n=11, we hav the value

=16.9009-10 x .04241

=16.9009-4241

=16.4768.

Comparing the two processes, we see that when we hav the values ovμ and 8 alreddy computed, Woolhouse's is decidedly the shorter. On the other hand, it is easy to see that Lubbock's formula applies, not only to annuities, but to other benefits; and that it will be applicabl to find the values ov such quantities as contingent annuities, the value ov which cannot be found exactly except by a very long series ov calculations (see Davies, p. 354).

I will next sho how the formula (3) may be applyd to find the value ov a temporary annuity-say ax.mn. Making the same substitutions as before, and observing that, since (3) would giv the value ov an annuity-due, we must subtract Vo (or 1) and add Vmn;

also, denoting the concluding differences ov Vo, Vn, Van,.... Vmn, by A'1, A2, A'3.

we get

n

1

ax.mn=n(Vo+Vn + Van + .... +Vmn)—1— (Vo+Vmn) 2

+ C1(A1—A ́1) — C2(A2 + A′2) + C3 (A3—A′3) — C4(A4+A′4) + . (9)

....

If in the calculation we stop at an even order ov differences, arriving, for instance, at a singl difference ▲2% ov the order 2k, we must consider that A'2k=A2k. In illustration ov this point I will first apply the formula to calculate the sum ov the 4th powers ov the natural numbers from 1 to 29. We hav, by processes which need no explanation, the folloing values and differences:—

Here n=7, A1=4,095, A'ı=473,025, &c.; and the sum ov the series 1+2+3+....+291, got by substitution in (9), omitting the term -1, is,

7 × 996259-3(1+707281)+ (4095-473025)

4

7

Mr. Woolhouse has givn (J. I. A. xi, 303) a formula by means ov which the sum ov the series can be found directly. It is equal 6 x 295+ 15 x 294 + 10 x 293-29

to

which givs 4,463,999, as

abov.

If, now, we wish to apply the formula (9) to find the value ov the temporary annuity, a40 28], by the same tabl as before, turning

to the figurs on page 314, we hav Vo+V+V14+ V21+V28= 2.8124; n=7, A1=—·2654, A'1=-·1407, &c., and the value ov the annuity is approximatly

7 × 2·8124—1—3(1 +·2079) — (2654–1407)

2

7

('0521+·0320) —·1808(0115-0086) —·1283(0029+0029)

The value calculated in the ordinary way is 14.9659.

Lastly, if we apply Woolhouse's formula to find the value ov

Taking the same exampl as before, we get,

040 287 x 2.8124-1-3(1+2079)

-404241-2079 (05700+02956)}

=14.9655.

On an application of the Theory of the Composition of Decremental Forces. By W. M. MAKEHAM, F.I.A.

THE determination of the probable effect, upon the increase of

population, of the extinction of small pox, is a problem which has been discussed at some length by three of the great continental mathematicians of the last century, namely, D. Bernouilli, D'Alembert, and Laplace. I propose, first, to solve the problem by what may be termed the theory of the composition of decremental forces, and then to reproduce (by way of comparison) from Mr. Todhunter's History of the Theory of Probabilities, the solutions given by the three eminent mathematicians in question.

The law of the composition of decremental forces may be thus stated:

1. Let Le denote the successive values of any given function

of the variable x, then the decremental force of the series will be

x

2. If F' denote the decremental force of a given series L'x, and F' that of any other given series L', then Fr, the decremental force of a series, L., subject to the combined simultaneous action of the two decremental forces F' and F', is equal to the sum of the two last mentioned decremental forces. That is to say, Fr=F'x+F'.*

3. From the last equation it follows that L=L'L', assuming that to the radix of each series (namely, Lo, L'o', and L'o,) may be given any value we please, by multiplying each term of the series by a constant.*

Let L' denote the number of persons surviving at age x out of a given number, L'o, born, taking mortality caused by small pox only into account; and similarly let L' denote the same with respect to all other causes. Then L, the number surviving in a body subject to all causes of mortality, is equal to L'x L'x.

Hence

if we form, from hypothesis or observation, the two series denoted by L and L' respectively, we may determine the series L ́r (which represents the effect of the extinction of small pox) by the equation

D'Alembert's statement and solution of the problem above referred to are as follows:-Suppose a large number of infants born nearly at the same epoch; let y represent the number alive at a certain time; let u represent the number who have died (during the period) of small pox; let z represent the number who would have been alive if small pox did not exist: required ≈ in terms of y and u.

Let dz denote the decrement of z in a small time, dy the decrement of y in the same time. If we supposed the z individuals to be subject to small pox, we should have,

But we must subtract from this value of dz the decrement arising from small pox, to which the z individuals are by hypothesis not

This expression may be easily reduced to that previously obtained; for y and z correspond to L and L' respectively, while

du

y

x

is equivalent to F'adx, where F'x is the force of mortality (or

du

=

decremental force) of the series L'. Hence eeFdz

D'Alembert appears to have been but little satisfied with the solution which he had arrived at. In a later publication, where he again touches upon the subject, he says:-"S'il est quelqu'un à qui la solution de ce problème soit réservée, ce ne sera sûrement pas à ceux qui la croiront facile." It appears however from Mr. Todhunter's work, that Laplace, in one of his investigations of the problem, gives the result in the same form as that deduced by D'Alembert.

Lz

I' x

The equation L'= is the complete solution of the general problem; for the total force of mortality is thereby resolved into its two elementary component parts, namely, that arising from the specified cause alone, and that arising from all other causes together. In the particular instance under consideration however, our knowledge of one of the characteristics of the specified disease, namely, the immunity from a repetition of the attack enjoyed by those who recover, suggests the possibility of resolving the function L' into yet simpler elements.

In the preceding investigation F' denoted the force of mortality from small pox measured upon L'a, the whole body of survivors at age x. Let F' denote the force of mortality measured upon that portion only who are still liable to it, namely, those who have not already suffered from the disease. And let L represent the body in which the annual decrement consists exclusively of the number attacked during the year by small pox. Then the number of

*If Fx denote the decremental force of any series Lx, the relation between them is universally represented by the equation e-Fxdx = Lx •