CONNECTED WITH HUMAN MOETALITY. 
519 
W also commences of significance at birth, but sinks gradually, and sinks into insigni- 
ficance at the age of 20, and then and after becomes of so small value that the terms sink 
into entire insignificance; the terms and — K) are values of significance, 
from birth to extreme old age ; the term is insignificant till x is equal about 80, 
“but for analytical anticipation is of significance some years before,” and then it slowly 
increases during the remainder of life*. The value is particularly interesting, because 
in the Carlisle Table it shows a surprising agreement with the stated mortality of children 
from the age of birth till the age of 1 year. 
Art. 7. But now continuing with respect to the original formula of near proximity 
to the law of mortality L^=<7.^| , where g, and q may for a long period be considered 
as constant, their values being dependent on the three selections of ages, and putting 
it in the form . q^ , and using a new and useful notation with respect 
to logarithms, by writing underneath a letter whose logarithm is to be expressed the 
prostrate small I, thus q^ to denote the common logarithm of q, q to denote the Napierian 
logarithm of q^ and the prostrate I in the reverse position, thus q, to denote the number 
whose common logarithm is q, and q the number whose Napierian logarithm is §', we 
have evidently from the equation \'L^—\d-\-Xg.q‘, 
1 1 1 
where the coefficients of x and its successive powers converge, in consequence of the small- 
ness of the Napierian logarithm of q, which in the Carlisle mortality is about -029, so 
that a very few terms of the series would be required, even if, for instance, the age a were 
30, and we wished to know the value of XLgo- To use this theorem with advantage, we 
should be provided with a Table of the values of g .(f and of d for every value of a, the 
youngest life of the three selected lives which are the foundation of the values of d, g^ q, 
their values being different according to the ages of selection ; and if 
&c. be, for the sake of example, as it was found by the above method, represented 
respectively by the converging series 
&c., &c., C-fa^-f ^C^^+^C^^ &c., 
it is evident on taking .r=0, that &c. will be represented by A, B, C, &c. ; 
and as the logarithms of the chances of the persons of the present ages a, b, c. See. living 
X years are respectively X logarithm of the chance of a living x years, 
of h living in x years, of c, &c. will be respectively 
^Kx-{-^Ax^-\-^Axf, Sec., *B^-f W+ W, &c., 'C.r+^C^^-f W, &c. ; 
and if r be the value of unity discounted for one year, using r to express its common 
logarithm, we shall have the logarithm of the present value of unity to be received in 
* But ft, though considered now to be constant, may show after x is 100 that it is not absolutely so, 
though data are wanting to show the nature of its variability. 
MDCCCLXII. 4 B 
