COKIJECTED WITH HIJMAH MOETALITT. 
515 
above, it is not ; and it is never in fact true if g and g are to be considered absolutely 
constant ; as is evident from the example of the method I used in my investigation, by 
what I call the vital rule of three. And as the Table from which the data are to be 
obtained is moreover likely to contain irregularities, and undoubtedly does contain such, 
which are not contained in the real law of mortality, it follows that if even the equation 
L^=d.yf , where d, g, g are supposed to be constant, did exactly express the real law of 
mortality, the values given to g and g by this vital rule of three would not be exactly 
the same for every selection, and therefore I should not have expected that any tolerable 
mathematician in reading my paper could suppose that I meant to state that I had given 
their exact values. But wherever the three lives were selected, had the Table been accu- 
rate from whence I selected the data, and were the law for constant values for d, g, g 
consistent throughout with the real law of mortality, their respective values would have 
come out the same. 
Art. 4. But neither of these requisites is to be depended on, as is proved by the 
two formulae given in my paper of 1825 for the Carlisle mortality, Art. 10 ; the one 
where the vital rule of three is based on the selection of the ages 20, 40, and 60, and 
the other where it is based on the ages 40, 60, and 100. The first of these equations is 
?vL,=3-88631-7v"'(2-75536+-0126^), 
or its equal, 
aL^=3-88631-~?i~’(' 0126^— 1-24464), or very nearly 
3-88631-X-'-0126(a;-100); 
"k standing for the common logarithm of, and the reverse, or the number whose 
common logarithm is &c., and the other, namely, from the selection of the ages 40, 60, 
100, gives the equation 
AL,=:3-79657-;^-X3-7467+-02706.r) 
:=3-79657-X-'(-02706a;-2-2533). 
And when the middle age of the three selections above is 40, we have 
Xd=3-88631, >.^=-0120; A(-?;y)= -1-24464, ^ 
sufficiently near — 1-26; and when the middle age is 60, of the three selections, we 
have 3-79657, -02706; sufficiently near the double of the former value; and 
?^( — Ay)=— 2-2533, also equal to nearly the double value in the former case, but not 
exactly in that ratio, but nearly in the ratio of 2 to 1x^5 5ut still, considering the 
nature of the data which furnish the values, in order to follow the hints these numbers 
give, 1 am inclined to think but lightly of the small variation from that proportion, 
and to suppose that the law of mortality, instead of being A.L^=X<Z~A“'(X(— X^)-{-^.Ay), 
when d, y, g are constant, which we have proved are only apparently constant, though 
for a long time exhibiting a strong appearance of constancy, should be, according 
to the above hints, = — A), with the addition of some small formulee 
to be sought for, where A is a constant from birth to extreme old age, and Ad,, Ay, are 
