estimation of the value of life contingencies. 235 
may be considered the general practice, must often lead to 
error. 
It may be worth observing, that if L 1+ „’ L i + »’ L ,+«> &c - were 
L 1 L/ L* 
respectively equal toL a . 7 r” «, h b .w" 6,1^7 r” c , & c. what- 
ever positive value n might be, that is, if the living from the 
respective ages a, h, c, See., whilst the time increased in arith- 
metical progression form series in a geometrical progression, 
then would the present value of the periodic income on the 
joint lives of the ages a,b,c,8zc. be the same for the same 
term, as on ages older than those by anv number of years, 
either the same for each, or different. And hence we may 
have some reason to suspect that the value of annuities given 
by tables on old age, by assuming a necessary term to life, 
as is done in the adopted tables of mortality, is likely to be 
far from the truth. 
Art. 5. As to the calculations of the values of n 
m 
a,b,c,&ce. 
r 
a, b, c, See. : a', b', c f , &c. : &c. 
See Art. 6 and 7. Sect. 1, and 
other expressions therein contained. Besides the usual mode 
of reducing them to their equivalents, a number of combina- 
tions of joint lives, for the purpose of working from calcu- 
lated tables, which, when the lives are many, will be a labo- 
rious task, and subject to the errors of numerous interpola- 
tions and of the other approximative modes which will be 
necessary in many cases when the lives are numerous, it may 
be easier to work them directly without such reduction, or 
by reducing the terms to geometrical progressions for short 
