[ 273 ] 
greater. Mr. Simpfon’s obfervations in this paffage, 
are true only, when applied to a different method ufed 
by himfelf in the 28th and following problems of 
his Treatife on the Dodxine of Annuities and Re- 
verfions. This method is exadt when the lives are 
equal} but it gives refults that are too far from the 
truth, when there is any confiderable inequality be- 
tween the lives. 
It is with reludtance I have made feme of thefe 
remarks. M. de Moivre has made very important 
improvements in this branch of fcience, and the 
higheft refpedt is due to his name and authority. 
This, however, only renders thefe remarks more ne- 
ceffary *. 
* The flri& demonftration of the folution I have given of the 
queftions here flated, is as fallows. Tt is plain that the pur- 
chafer of A’s right, as flated in the firftqueftion, cannot get into 
pofl'effton till the year when A and B (hall be both dead, nor 
then, unlefs A happens to die lait. Now, fuppofing the com- 
mon complement of life », the probability that A and B fhall 
be both dead at the end of the firft year, and A die laft, is 
I 
n — x 
X I 
n 
n — 1 
n 
n — 1 , n — rl 
2 n 2n- * 
In 
like manner, the probability that they fhall be both dead at the 
end of the fecond and third, Sic. years, and A furvive, is \ — 1 
&c. The 
prefent value, therefore, of the ift, 2d, and 3d, Sic. rents of 
, - „ . 1 — I n~- \ n— il“ 1 
the reveruonary ettate, is — — , ■ — ? 
2r 2 nr znr 2nr~r J z> 
n- 
“x + 
n— 2I 1 
«— 3 
2 nr" znr 1 " ' 2 «V* * 2 r 3 2«r J 2 nr* ' 2 n“r i 
r fignifying 1 /. increafed by its intereft for a year ; and the 
efiate being 1 /. per annum. And the fum of thele terms is the 
Vol. LX. N n la 
