Survivor/hips. 275 
from the different feries, and is known to be accurately true 
from felf-evident principles. Q. E. D* 
I have now given general rules for determining the values of 
reverfions depending upon three lives in every cafe which, as 
far as I can difcover, will admit of an exaft folution. The 
remaining cafes, which are nearly equal in number to thofe I 
have inveftigated, involve a contingency for which it appears 
very difficult to find fuch a general expreflion as (hall not 
render the rules much too complicated and laborious. The 
contingency to which I refer is that of one life s failing after 
another in any given time. The fractions expreffing this proba- 
bility are every year increafing, fo that the value of the rever- 
fion muff be reprefented by as many feries at leaffc as are equal 
to the difference between the age of one of the lives, and that 
of the oldefl life in the table of obfervations. 1 have indeed fo 
far fucceeded in the method of approximation as that the rever- 
fioti may be generally afcertained within about J^th part of its 
exad value ; but I fhall not trouble the Royal Society at 
prefent with th.efe inveftigations. 
The 54th, 35th, and 56th problems in Mr. Simpson’s Se- 
lect Exercifes involve this contingency, and, by the afliftance 
of M. de Moivre’s hypothefis, admit of an eafy folution. 
But fuch is the fallacy of this hypothefis, that it renders Mr. 
Simpson's conclufions ohvioufly wrong, though his reafoning 
is perfedly corredt : for it cannot finely be an equal chance in 
all cafes that one life fhall die after another. In the fhort 
term of a fingle year the chances are indeed fo nearly equal, 
that it would be wrong to perplex the folution by attempting 
O a z greater 
1 
