estimation of the value of life contingencies. 251 
±j , 
tn = p+ v , — ir^-) Moreover, though the chance of A’s 
a 
surviving B, being half the chance of their both being dead, 
is here derived from the hypothesis of the decrement's being 
proportional to the time in each original age ; this is not 
the only hypothesis which will cause that relation of the con- 
tingencies ; or, which is the same thing, provided they both 
are to be dead, that the contingency of A's surviving B shall 
be equal to the contingency of B's surviving A ; for the 
fluxion of this equation is — (1 - ■ 
^a+x 
"a+x 
J b+x 
(1- 
2±a 
L * ) 
— L 
b+x 
L L L, 
x —jf ’ ; and therefore l — l ~— — > and consequently 
£ a+x b + x 
L I Li . \ 
taking the correct fluent &c. we get 1 j—-— K . (1 ); 
k be a constant quantity; therefore the fluent of (1 ^ 
L, 
—if ; which taken to 
.-iif = — the fluent of k ( 1 ~ x 
vanish when x = 0 is the contingency according to the pre- 
sent hypothesis of B dying after A ; and is = & . (1- 
J b+x 
iii / ^b + x\ 
= 2 -k.(i. 
.2^+(f£±iYW-* . 
(1- 
J b+x 
17 
— . 1 
2 
J a+x 
J b+x 
Hence it appears, that whatever the decrements of life, or 
the constitution of the functions of life, it will be an equal 
