CONNECTED WITH HUMAN MOETALITT. 
539 
call this 
&c., 
then on multiplying minus the fluxion of this by the anti-Napierian logarithm of the 
sum of all logarithms of the separate chances of each life in the combination C being 
living, increased by (previously § being the Napierian logarithm of the value of unity 
certain due in one year at the rate of interest to be involved in the calculation), the 
fluent of this will give the value required, and will not require many terms of the result- 
ing series, which I represent by 
f) + f ) , &c. , 
t being the period of commencing of the assurance. 
Art. 20. And much more intricate cases may occur in valuations of property offered to 
the public and to reversionary companies. But instead of in this place dwelling on the 
intricacies which may occur, I will refer to art. 5 of my paper of 1820 with respect to 
calculations referring to the formulae of the convenient notation there used,. 
«, h, c, See., 
a, 1), c, See., d, V, c'. See., 
in which I have developed the various combinations and their connexions of annuities 
of 1, 2, 3, &c. joint lives which would come into the calculation, which I derived by 
the development of the expression 
„) X „+!)*, „) X „) X &c.=l, 
when x=l, where E„ „, Sec. denote the chances of persons of the ages a, h. See. 
being living in n years, and „ the chances of their being then dead ; and in 
consequence 
E5_„-|-D4^„=1, (fee.. 
the X attached to the E being supposed =1, but only introduced, as w, to point out, by 
its exponent in the development, the various combinations of persons on which the 
annuities are to be calculated, if, previously to the development, „ be expunged 
by the equations 
E„,„, Ea,„, &c. 
But had we introduced y, also a representative of 1, and written the expression 
xE,^ „ X aE 4 ,„+^Dj, „ x &c. = 1, 
and not expunged D, the exponents of x and y in the developments would represent the 
combinations and connexions of the various cases which Avould occur of persons living 
and persons dead in connexion with each other in the time n ; so that if we chose to have 
Tables constructed on a certain number of the persons being dead of the whole number, 
we might have a variety of modes of calculating the value of life contingencies, instead of 
being dependent on the value of joint lives; but I do not mention this by any means as 
a recommendation, but quite the contrary. But if Tables of the reversions of annuities 
