534 
ME. B. GOMPEETZ ON THE SCIENCE 
me very satisfactory; but I do not repeat that Table here, as I have given a Table 
from what I consider the improved formula : but I gave in that paper the values for the 
four cases, namely, Carlisle, Northampton, Sweden, and De Parcieux, of the aforesaid 
valuable constants of the formula just stated, and there alluded to ; in this I mean to 
subjoin the results for tables of mortality of constants for the last three places, which 
are satisfactory in my opinion. But I am not able to say if, before this goes to press, 
I shall be able to give the results which an investigation of the constants in the for- 
mula, which I consider an improvement of the other, wiU give for the last three morta- 
lities. 
Art. 15. The term ‘expectation of life’ in common use is not applied to that which I 
should call the orthodox expectation of life, and therefore, not venturing to discard the 
usual adoption of it, I will introduce the term orthodox expectation of life for that term of 
years to which it is an equal chance whether a person of a given age shall reach or not ; 
and I observe, if the Napierian logarithm of the chance of a person of a given age living 
X years beyond that age be represented by which will be a nega- 
tive quantity, this must be put = — *691472, namely, Napierian logarithm of -J. And the 
value of X which will result from that equation is that which I call the orthodox expec- 
tation of life of that person; and if there be any number of joint lives, and the sum of 
the Napierian logarithms of the chances of each separately living x years be repre- 
sented by ^Ax-\-^Acif-\-'^Kx^-{-Scc.^ if this be put = — *691472, namely, the Napierian 
logarithm of the value of x, which this equation will give, will be the period 
beyond the present to which the joint existence of those joint lives has an equal 
chance of attaining or not attaining ; and as the coefficients ’A, ^A, ®A are very con- 
verging, a very few terms will be sufficient, perhaps merely the first term ; and if there 
be two separate combinations of joint lives, which I will call the A combination and 
the B combination, and the Napierian logarithm of the chance of the A combination 
lasting X years be represented by ^Ax-{-^Kcif-{-^Kx^-\-Scc., and that of the B combina- 
tion lasting X years be represented by 'B^-f &c., then if these two be equal 
to each other, and x comes out positive and not beyond the limit of the accuracy of the 
theorem, x will be the term to which it is an equal chance of one combination in parti- 
cular surviving or not surviving the other ; but should x come out negative, that term of 
years does not exist within the limits at least of accuracy of the theorem. 
Art. 16. If there be two sets of lives, which I will call the A combination and the B 
combmation, and the separate chances of their existing x years be respectively repre- 
sented by 
l-{-^Ax-\-'^Ax^-\-^Ax^-\-Scc,, and l-j-'B.r+^B^“4-^Ba;®+&c. 
respectively, then the chance that the B combination shall fail during the continuance 
of the A combination, and between the periods t=- a given quantity to t=x, will be the 
fluent of 
&:c.)x —fluxion of (‘B.z'-l-^Ba^-l-^Ba^^ &c.} 
between those limits, because the fluxion of the discontinuance is minus the fluxion of 
