contract, or departure from West Africa during the period of the investigation.

Under certain conditions it might be considered advisable to limit the period of observations, as, for example, when it was known that in the country under investigation the rates of mortality had improved considerably of recent years.

The age at commencement would be the nearest age at the date of commencement of extra risk. The duration in the case of the deaths would be the number of complete years lived. In the case of the existing at the closure of observations and the withdrawals, the duration would be the exact duration correct to one decimal place. If the data would permit of it, it would be convenient to trace the existing only up to the anniversary of the date of commencement of the extra risk in the year of closing the observations. This artifice would secure an integral exposure in the case of the existing, and thus lessen the work of the investigation. It will, however, generally be found that the data is limited, and on this account it is advisable to close the experience upon a fixed date.

If the object is to find the rate of mortality according to age the first arrangement of the cards would be into groups of ages at commencement of risk. It might, however, be considered sufficient to investigate the rates only according to duration, and for this purpose to combine the experience of all the ages in one table.

The cards would then be arranged into Existing, Discontinued, and Died. Each of these groups would be arranged according to the duration of the policy, and their numbers counted and tabulated in a Schedule of the following form. According to which of the methods above indicated the table is being formed, the heading would be:—

AGES AT COMMENCEMENT, ≈ TO INCLUSIVE, OR ALL AGES.

The Exposed to Risk during the (t + 1)th year of duration

It must, however, be kept in mind that the rates of mortality thus deduced are only in respect of the deaths which occurred while the lives remained in the West Coast of Africa, and relate only to the period of the investigation. No provision is made for the fact that a proportion of the Existing and Discontinued will be damaged lives. In other words, there is no discrimination between the lives who might be said to have left

the West Coast of Africa to die at home and those who left in the full vigour of health. This point would require to be carefully borne in mind when fixing upon rates of extra premium to be charged for such risks.

POPULATION STATISTICS.

If a population could be shown to be in all respects stationary, the number of births each year being equal to the deaths, all the births taking place on the same day of the year, say the 1st January, and the community remaining undisturbed by either emigration or immigration, it is obvious that a census of the population on any 1st January would give us a mortality table based upon population statistics. The number of births, termed lo, would be the radix of the table, the number living at age 1 would be the survivors to that age out of the , persons born one year ago, the number living at age 2 would be the survivors to that age out of the l persons born two years ago, and so on. Or again, if we arranged the deaths during any calendar year according to age last birthday at date of death, we would have a mortality table. The deaths between 0 and 1 would have occurred out of the 1, persons born at the commencement of the calendar year, the deaths between 1 and 2 would have occurred out of the l births two years ago, and so on. The total of the deaths would be equal to the births, and the continuous deduction of the deaths during each age interval from the number living at the commencement of the interval, would give us the number living column.

It is obvious that a population of the nature described does not exist. It is more in accordance with experience to assume merely that the births and deaths are uniformly distributed throughout the year, and bear no fixed relation to each other.

The definition of the function usually termed the central death-rate contains within itself a method of forming a mortality table from census statistics. If we make an enumeration of the population according to age attained on the 1st July of any calendar year (that is, include every person who at that date has passed his æth birthday, and who is in his (x + 1)th year among the L or number living between precise ages x and (x + 1)), and if we arrange the deaths (d) during the same calendar year according to age (x) last birthday, at date of death, then for any year of age the division of the deaths by the number living will give us the central deathrate. For the age interval x to (x+1), this function is denoted by the symbol mx

In accordance with the definition, we have

The ordinary probabilities of death (4) or survivorship (pa) are always expressed in terms of the numbers attaining precise ages. It is therefore necessary in order to obtain p and q to transform the L to a function of 1. It is obvious that on the assumption of a uniform distribution of deaths throughout the year, L will be equal to the mean of those attaining precise age x, and of those attaining precise age (x + 1).

Making the assumption of a uniform distribution of births and deaths throughout the year we have

Having obtained our table of me and graduated the same, we would pass therefrom to a table of Px by means of this formula. Assume a radix of say 100,000 at age 0, i.e. l = 100,000. Continuous multiplication of the radix by the successive values of p will then give us the mortality table, since

lo × Po = l1, l1 × P1 = l2, l2 × P2 = l3, etc.

One year is, however, too short a period upon which to base a mortality table. The data would of necessity be limited. There might, moreover, be special influences at work affecting the rates of mortality during the whole or a portion of the table, such as an epidemic, say the prevalence of influenza, which would chiefly affect those advanced in years.

If we extend the period of investigation so as to include say two years instead of six months on either side of the date of the census, a more solid basis would be obtained. It would still, however, be advisable to compare the death-rate per 1000 in each of the years forming the period with the corresponding ratio of other years, so as to ensure that the period selected for investigation was average as to mortality. Unless the migration element should be found to be fairly uniform throughout the four years, a correction might be necessary on this account.

In the census returns of this country, the number living is given for each year of age during the first five years, in quinquennial groups from ages 5 to 25, in decennial groups from ages 25 to 85, and for ages over 85 is given in one total. A "graduated total of the estimated population" is also supplied, but no particulars are given as to how the figures are obtained.

Thus we are able to obtain from the census statistics the numbers living at the date of the census, arranged for the most part in quinquennial groups of ages, and from these data we have to obtain the denominator of

From these figures we are able to obtain approximately, as will be shown, what might be called one year's population.

The data for the deaths can be obtained from the annual returns of the Registrar-General.

The census of this country is made decennially on the 31st March, the latest enumeration having been made upon the 31st March 1901. To get the deaths for two years on either side of this date, the assumption of a uniform distribution of deaths throughout each calendar year might be made,

and the deaths during the four years' period assumed to be three-fourths of the 1899, the whole of the 1900, 1901 and 1902, and one-fourth of the 1903 deaths. The sum of the deaths during these four years would require to be divided by four, in order that one year's population might be compared with one year's deaths.

The data would consist of the following:

TABLE I.-Population in Age Groups at 31st March 1901.

TABLE II.-One-fourth of the Deaths during the period 31st March 1899 to
31st March 1903 arranged in Groups.

Given the deaths and population arranged as above, it will be necessary to employ some method of interpolation to enable us to pass therefrom to the corresponding yearly groups. The method employed by Joshua Milne in the construction of the Carlisle Table is that usually adopted. Crossruled paper, of sufficient size to admit of the accurate reading of the ordinates, would be used for the purpose.

The base line (or abscissa) represents the age attained, while the upright lines (or ordinates) represent the number living. The altitude of the first rectangle will be the number living in the age-group 5-10 divided by five, the altitude of the second rectangle the number living in the agegroup 10-15 divided by five, and so on. A smooth curve is then drawn proceeding through the several rectangles, care being taken that whatever is added on to a rectangle is likewise in equivalent cut off. Each rectangle is then divided into yearly intervals. These yearly intervals are bisected, and the ordinates erected at these points produced to meet the curve will represent the number living in that age interval. The sum of the ordinates of any group must of necessity be equal to the number living in the group. What has been done, in fact, in the case of a quinquennial group is that five figures have been constructed, bounded on the top by an ostensibly

curved line, the area of the five figures being equal to the area of the rectangle. The deaths will be treated in a similar manner.

Having thus obtained complete tables of Le and d, the division of the latter by the former will give us a table of the central death-rate. This function should then be graduated to remove any accidental irregularities. From a table of graduated me will, as previously described, be obtained a table of Px.

The application of Milne's method to the period of infancy is not satisfactory. Professor Pell has devised a means of obtaining the early rates of mortality from the number of births as given in the returns of the Registrar-General and the deaths as registered. The deaths between 0 and 1, during the calendar years 1901, 1902, 1903, and 1904 would be compared with one-half the 1900, the whole of the 1901, 1902, 1903, and half the 1904 births. This would give us q, the probability of a child dying in the first year after birth. The deaths between 1 and 2 during the same period would be compared with one-half the 1899, the whole of the 1900, 1901, 1902, and half the 1903 births. This would give us 1/4, the probability

9 10 17 12 13 11, 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

that a child aged 0 will survive one year and die in the next. But 1/%=Po 91 We have already found go and po=(1-9), whence we can obtain q1. Again, the deaths between 2 and 3 in the same period would be compared with one-half the 1898, the whole of the 1899, 1900, 1901, and half the 1902 births. This would give us / Po x P1 92, whence 2 is = × X obtained, and p2 = (1-2). Similarly would be obtained 3/10 - Pox P1 × P2 × 3 and / Po x P1 P2 P3 4. From these equations p, and × p1 would be obtained. We have thus found po to P, inclusive. These values would require to be graduated with reference to P5, Pe, etc.

X

Having assumed a radix at age 0, we would, by continuously multiplying by the successive probabilities of living, obtain the life table.

Another method of obtaining the statistics is that followed by Milne himself in the construction of the Carlisle Table already referred to. His plan was to take the number living on 31st March 1891 and 1901, and the deaths during the ten years' interval. The numbers living in the various age groups at each of these dates were added together and multiplied by five, in order to put the population and the deaths on a common basis for