The problem is as follows. Given logg, log q, a=loges, d=log€(1+1), it is required to find the annuity on a given combination of lives. I give the solution for one life; in the case of k lives, aged x, x+k1, x+k2, &c., the solution is the same, except that for a we must substitute k.a, and for g, g′ = g1+qk1+qk2+....... The characteristic of log T1+r is —1, or 9.

By way of illustration, I take the two examples given by Mr. Makeham (J.I.A., xvii, 309, 310), taking his values of n and z, and deriving from them r and v by the relations,

r=1-M.n,
loge"=10%.

In these cases the working out of the formula would, using 5-place logarithms, be as follows:

These examples would, as given, be unintelligible except in connection with Mr. Makeham's paper. The basing of the work on the quantities n and z is of course unnecessary, and has only been done on this occasion to show the connection of the two methods. The following is a new and distinct example of the present method, beginning at the very beginning.

In the HMF tables published by Mr. Brown (J.I.A., xvi, 428), the constants are, log g=-00041, log q=04, log s='00286; required the 3 per cent. continuous annuity at age 40. Here 8=029559, loges='006585=a, and a +8='036144.

The demonstration of the formula remains to be given. Mr. Makeham has shown (J.I.A., xvii, 308) that

his symbol m being replaced by its equivalent, r-1. The following property of the gamma-function is well known:

VOL. XVIII.

fe ̄”.v".dv:
=n.com-do

S

By means of this property may easily be derived the modified

formula,

Multiplying this series by e-", and also by the expansion of e", its value remains unaltered, and we have

[„e ̄*. v*.dv=C—e ̄*.vl+r{

1

+

+

v2

11+r (1+r). (2+r) ̄ (1+r). (2+r).(3+r)

=C-e-v.vr-1.(Vg + Vg + V4 + ....).

+

If v=0, all of the second member vanishes except the constant, whence CT+=foe-.vr.dv. Dividing the equation now by e-v.v-1, we have

Substituting this result in the expression last found for ax, we obtain directly the desired formula,

r−t+v+vq+...

āx= r. (a+8)

The series for se=”.v.dv may also be obtained by integration by parts, or by applying Leibnitz's Theorem for the determination of the nth differential coefficient of the product of two functions, making n=-1.

If r<0, T1+r must be found by means of the well-known T2+r relation, Ti+r= If r, whether positive or negative, 1+r

approaches closely to zero, the nearer it does so, the smaller will be the number of decimal places to which the resulting annuity can be depended upon; and when r=0, the formula is reduced to 0

the form. In determining I from the table, interpolation by 0°

first differences will usually be sufficiently accurate; but towards the beginning of the table second differences will have to be resorted to.

In place of v, v2, v3, &c., may be substituted a corresponding series of powers of u="-1, thus :—

and so on. A series of powers of y=loge (1+v) might also be employed, but being less convergent than the others would be practically valueless. These series may be derived from the series of powers of v by expansion and substitution of v=loge (1+u) and v=-1, or they may be obtained in the same manner as the original formula, by transformation of the integral, expansion, integration, and multiplication.

The series of powers of u is less symmetrical and less convenient for calculation than that of v, but it is usually more convergent, requiring a smaller number of terms to give a close approximation to accuracy. On the whole, however, I am disposed to prefer the use of the powers of v. To illustrate the comparative convergence of the two series, it will suffice to quote the three examples already given.

On the Value of a complete Annuity when payable by m equal Instalments in each Year. By THOMAS CARR, F.I.A.

IF the death take place in the (t+1)th interval in the year of

death, then the present value of the t instalments of

1

m

each paid

1

the year, multiplying the latter formula by and summing it for

m

all values of t from 1 to m-1, both inclusive, we have

or putting log. (1+i)=8, whence 1+ies and i=e—1;

or using the well known development by the numbers of Bernoulli;

the day of death, is (see Journal, vol. xiii, page 363)

Adding the latter to (a) we have the value of the total sum paid under the annuity in the year of death