3*2 Mr . Morgan 
confiderable, efpecially if the expectations are derived from the 
London, the Sweden , or any other tables, in which the de- 
crements of life are unequal. But when three lives are in- 
volved in the queftion, thefe errors are generally enormous ; 
nor is it ever fafe, when the ages of thofe lives differ very 
much, to have recourfe to rules which are founded upon this 
principle. The three following problems, though the moil 
common in the doCtrine of furvivorfhips, have never hitherto 
been folved in a manner ftriCtly true. The fecond of them is 
of particular importance, and I have taken much pains to exa- 
mine how fir Mr. Simpson’s folution of it may be depended 
upon. It has, indeed, been folved by M. de Moivre *, and 
Mr. Dodson •f' : but the firft of thefe writers has erred moffc 
egregioufly in the folution itfelf, and the other having derived 
his rule from a wrong hypothecs, has rendered it of no ufe. 
It is much to be wifhed that the folutions of all cafes in rever- 
fions and furvivorfhips were deduced, like the three following 
ones, from the real probabilities of life. Moft of thofe which 
are now in ufe are at beft but approximations, and can never 
be relied on with any tolerable degree of fatisfatftion. 
PROBLEM 'I. 
Suppofing the ages of two perfons, A and B, to be given ; 
to determine the probabilities of furvivorfhip between them 
from any table of obfervations. 
SOLUTION. 
Let a reprefent the number of perfons living in the table at 
the age of A the younger of the two lives. Let a\ a'\ a'" 
* See Mr. de Moivre’s 17th problem, and Dr. Price’s remarks upon it in 
his Treat ife on Reverfionary Payments, EflTay 3. Vol. I. 
f See Dodson’s Mathematical Repofitory, Prob. 23. Vol, III. 
