COm^CTED WITH HUMAH MOETALITY. 
53T 
rities of the path to repay with gratitude the benefit they have received, and not in 
self-conceit to forget that a dwarf on a giant’s shoulders may see a more extended pro- 
spect of beauties than the giant could ; and hoping to be excused for this digression, I 
will proceed by observing that the best way of solving intricate questions relative to the 
value of life interest, may not at all be, as is usual, to have recourse to the value of life 
annuities. With respect to the expression given above, namely the fiuent of 
&c.) X — fluxion of + &c.), 
and the one which follows, I observe that if in the two sets of combinations of lives all 
the lives of the combination A were required to be living, and a specified number of 
specified persons of the combination B were also required to be living, and the assurance 
is to be on the failure of the remaining portion of joint lives in the B combination, pro- 
vided all the lives of the A combination and all the fives specified of the B combination 
were also living, then the solution would be the same, with the exception that all those 
specified fives of the B combination must first be transferred to the A combination ; this 
must be evident ; but it is mentioned that in case with respect to the B combination 
only a certain number specified, without specification of which lives of the B combination, 
are to be living, the question may be solved with this difierence, that the sum of all 
the values must be taken which will occur by the various modes of withdrawing the 
specified number from the B combinations; but there are cases where a less laborious 
mode may apply. 
Art. 19. But returning to the originally expressed combinations A and B, suppose it 
were required to find the value of an insurance of unity on the failure of the combina- 
tion B, provided it happened during the continuance of the combination A. Let the ex- 
pression &c., instead of representing, as before, the anti-Napierian 
logarithm of the sum of all the Napierian logarithms of the chances of existence of the 
different lives separately in the combination A, now represent the anti-Napierian loga- 
rithm of that sum, having the coefficient of the first power of x increased by the Na- 
pierian logarithm of the present value of unity due in one year at the rate of interest 
to be involved in the calculation, then the fluent of the expression 
(l+^A^-f=^A^+W, &c.)x(-'Bi;-2.^B^^i^-3. Wi;, &c.). 
instead of being the chance of a failure of the combination B between the given period 
and X during the continuance of the combination A, will be the present value of unity 
payable on that event taking place, and so of others. For instance, if only three lives of 
the present ages <?, b, c were concerned, then the value of the assurance on the life A, if 
it happened in the lifetime of B and C, which is the first of art. 6 in my paper of 1820, 
would be solved thus : supposing the present ages of A, B, C to be a, b, c, and the 
Napierian logarithms of 
Li ’ La 
to be respectively 
A,x+A.,af‘+A^x^+ &c., B^x-\-B.,x^-\-B^x^ &:c., 
QiiX-\-Qj<^x^-\-Qi^x^-\- &c., 
4 D 2 
