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Published online by Cambridge University Press: 18 August 2016
This paper had its origin a year or two ago in an investigation entered upon with an object which has since fallen into abeyance. At that time, it being desired to obtain a thoroughly well-graduated table of mortality giving a favourable view of human life, attention was directed to the Carlisle Table, and sundry attempts were made to remove its irregularities.
Many of the experiments then tried would not now be of any public interest, but the effort to use Mr. Makeham's formula brought out a few facts which it is perhaps worth while to record; and, at any rate, whatever may be thought of the present notes, the authors do not consider themselves to be guilty of any impropriety in bringing forward for discussion once more the practical employment of Mr. Makeham's expression for the law of human mortality.
They are the more emboldened to suggest this subject for debate, in that Mr. Walford, at a late meeting, announced that the formula was found in America to give perfectly satisfactory results when applied to a table formed from the experience of assured lives on that continent.
The first step in the original investigation was to graduate the Carlisle Table (as published by Mr. Milne) by means of Mr. Woolhouse's formula; and as the resulting curve was not entirely free from roughness, the process was repeated. The changes introduced by the second graduation were insignificant, but the table produced was one of very great smoothness, although there was a wave-like form observable in the succession of the values of some of the functions.
page 192 note * It will be remembered that Dr. Farr symbolizes by mx the ratio between the recorded deaths in the year of age x to (x +1), and the enumerated population in the same year of life. On the assumption that in the life table this ratio will be equal to dx ÷lx+½, he then forms the column lx . The ratio is approximately be equal to mx is approximately equal to μ x+½.
page 195 note * When the paper was read at the Institute, we were under the impression conveyed in the text; but in the discussion, Mr. Sutton pointed out that we had done Mr. Woolhouse an injustice. The final table was not formed by assuming a mean of constants, but by assuming a mean for log q, log l 10, Δlog l 10 Δ2log l 10; and and by calculating in the usual way the final constants from these l1 0 values. Mr. Woolhouse, unfortunately, has not described his process very lucidly, and others beside ourselves have misunderstood him. It is almost a natural inference from his explanation, that he took a mean of constants ; and if a reader come to that conclusion, and test it, he will find his opinion confirmed, because two of the four final constants of Mr. Woolhouse are exactly means of the preliminary constants, and the other two differ from the mean by only 3 in the seventh decimal place.
