CONNECTED WITH HUMAN MOETALITY. 
525 
But the formula I now state is more convenient for ultimate reduction ; but the difference 
of the two being while stands for the common logarithm of, and -=> for the anti-common 
logarithm of, in the formula referred to, the values of C, are not the same in this 
as in that, but would be, if multiplied by the Napierian logarithm of 10 ; and I will begin 
by showing the great use which even the original formula 'L^=zd.g{‘'" may be, though 
d, g, q are, instead of being absolute constants, quantities very slowly variable from one 
selection of three lives to another ; though it is not equally valuable in point of accuracy 
to the improved formula, where all the quantities except x are constant from birth to 
extreme old age. And now, reverting to the formula \j^=d.g\ , and observing that in 
the Carlisle mortality the selection of the three ages may be distant from each other by 
even 30 years, being 10, 40, 70, we obtain a very efficiently-useful formula, although in 
some cases, though rarely, given by the formula, and of Milne, may even differ 
two years ; but still, the proportion of the chances of living to those ages given by each 
will be but a small per cent, of each other. When the selection is 10, 40, 70, we have 
a^d^ the Napierian logarithm of (^=8’8833; <^q or the Napierian logarithm of 
(?=-0377355; ^=--0786136 ; -(-.^)=2-89605 ; -0^^2-57675. Here the difference 
of ages in the three selected lives is successively 30 years ; but as that difference may 
not give sufficient accuracy, I do not adopt it. 
If , with constant values of d, g^ q, were true throughout life, we should 
have the logarithm of L^=logarithm of logarithm of g.(f, and 
XLa+^=logarithm of logarithm of g\-q''-q^-, 
as before observed; and the logarithm of the chance of a person of the age a being 
living in x years = the logarithm of logarithm of g.q\q^—l)-, and similarly the 
logarithm of the chances of persons of the ages h, &c. living x years will be expressed, 
logarithm of^.^*(^— 1), &c. ; and consequently we shall have the logarithm of the 
chance of persons now of the ages a, b, c, e living x years 
= logarithm of gx{q“^—l){q''-{-q^-\-q^-\-q'^), 
and also the logarithm of the chance of a person of the age qt living x years 
= logarithm of gX(q^ — V).qP-, and therefore if ^ be found so that q^=q'^-\-q’’-\-q‘'-\-c/, 
or, which is the same thing, if p be the value of + g‘' + ^ 
will the chance of a person of the age p living x years, and the chance the four per- 
sons of the ages a, b, c, e being every one surviving in x years, be exactly the same. 
This circumstance had induced the learned Professor Augustus De Moegan, when 
commenting on the theorem lj^=d.g\ , which was given by me in the Philosophical 
Transactions in the year 1825, and the ingenious Mr. Speague, and others who appre- 
ciated it, and philosophically felt pleasure in bestoiving praise where they thought merit 
was due for the discovery of a useful theorem, with good analytical judgment to observe, 
