Mr. Gompertz’s analysis applicable to the 
242 
7 L r> q + -7 L q,r- f 7 L 7+Ej r +7 L ?+£ , r+£ ; and consequently 
2 
or 
0 . 
0 
* + f 
T 1 T I I T 1 T 
Lq+x ‘ r + a? 2 7, r z 2 ^q + t, r "1" 
T L , + „r+.+.-— T L «+.+.',r t , + ;l (t , w , +lV , whereas if 
in Art. i, we take and ^rrg-f-e', we shall have 
.T 
O 
o 
E + E ' 
-j-x ■ L V +ir 
2 r 2 r + £ + e< 2 ^9 + £ + s 1 ~ 1 ~ 
j 
- L fl + g + lf r + s + E '. And we are here presented with two dif- 
Lq+x . L r +x’ that is of the 
ferent forms for the value of 0 
s+/ 
value of the fluent of L q+x , . L r+x , whilst x from o becomes 
s+s' ; the latter form being on the supposition, that the de- 
crements are proportional to the time throughout the inter- 
val e-j-s', and the former, that they are first proportional to 
the time during the interval e, and that after that they are 
again proportional to the time through the next interval e\ 
The reason of adducing these two, which are commonly ap- 
proximate values of 
. L / . +a? , is that several ingenious 
authors in the solution of some of the problems hereafter to 
be considered, have taken as an approximation that if two 
persons, who were living at a particular period, are both dead 
at any nominated period after, within a certain limit, that it is 
an equal chance which of the two is the survivor ; which as 
will appear is the natural consequence of supposing the de- 
crements of the lives during every part of that period, to be 
in a constant proportion to the time elapsed from the com- 
