8 An algebraical Expression of the Values of Lives. 
it would otherwise express the value of the continued payment 
at the time of birth, instead of the actual present value. 
The same solution may be applied to the expressions for the 
value of joint lives, multiplying always the product of the survi- 
vors by v*; the fluents being all comprehended in the general 
n(u—=1)) "eo 
. 1 a 
theorem /utx"@ = v7( — x” — ieee oa 
HLU (sLv)2 (HLv)? 
n{n—1) (rn—2) 
({uLv)? A 
three or more lives, the quantities m, 4, and z, may be determined 
from the theoretical mean values of each, or otherwise, and the 
present value, thus approximated, will differ much less from the 
truth than the theoretical mean values would do, if originally 
found in a similar manner. The usual deduction must be made 
for periodical payments, whether annual or quarterly: and a 
slight correction may sometimes be required, on account of the 
different effects of annual and continual interest ; but both these 
points are very easily arranged: commonly, indeed, the subtrac- 
tion of half a payment from the present value, thus computed, 
will give us the corrected value with considerable accuracy. 
crag he ae ote For the joint present value of 
The great labour required for Mortality of London 1815. 
such calculations, according to SURVSVORS 19560. AGE. 
the usual methods, renders it very Ne SERS ST fd 
difficult to adapt the tables of an-~ i 
nuities to every possible variation sos Sree ee 
of the value of life; the improved : 
habits..of, society,:and ‘probably’ “~ 9 . |._-298 
also the advancement of the me- 
dical sciences, and especially the 
introduction of vaccination, seem 
to have effected by degrees a very 
material change in the mortality 
of this metropolis ; and it appears 
that the magnitude of such a 
change, and its operation, in its 
various modifications, may In many 
cases be much more conveniently 
appreciated by this mode of find- 
ing a continuous law for the de- 
crements of life, than by the in- 
spection of tables, and the num- 
berless combinations of their ele- 
ments, 
