estimation of the value of life contingencies. 271 
constant y- : a - - -j fluent of 
fluent of 
J a, b 
“'c-J-x ’ ^b + x 
^“c+x ' ^a + x 
L ~"L 
4 “ 
CL ~f" X 
; we find this value under the idea of 
constant decrements for the whole term to be 1 — 4- 
.(— * - +i ~ ■ 
V nr. I z an 
b 
* L l± 
be 
J c'fl _L I «'c'\ 1 «' ( 11 
a ac 3 ac I 4" a V J ^ Ti 
1 fl/5V j_il £ f __ jl * li'_ L — 
1 be * 2 b . c ) * abc 2 b 2 a 2 be 2 be ", 
and under the same hypothesis of the constant decre- 
ments, during the whole time n- f-i, we have the corre- 
n" h e r" b n n u h 1 ' r" 
spending contingency i + 
— \ d-L ; and the excess of this above the contingency for the 
be 
. t b'—h" , t a'-a" , b'c'—b" c " , a'c'—a" c" a' b' d —a" b" c " 
term is \— i — Iri— — f- 
2b c 
4 - 
2 a c 
2 a c 
the same except in notation as before, as it should be ; the 
difference of the operation only being in the notation. But if 
we only consider the uniform decrements to reach to the 
period n , and then during the next year take proportional 
decrements during the year, we shall have the fluent of 
l , . L . 
c- frx a + x . r cic t T 
— L . l ' answering to.r = w + 1, = — -f- - 4 - £ 
ca’ 
a c 
C a 
a g 
-j- — t ~ 4 “ See Art. 6, Section 3; consequently 
Lj L c+x L a+x 
the value of — — 1 . fluent of — answering to x=n 4-1 
a ’ c“ 
L c ' L b 
IS = 
b" 
'b" 
1 L JL “ __ JL 
3 b » 3 ab 
i_c / b" , b"c'a" , a'b"c" 
+ j_ u v u _ i_ 
2 n h P 3 
ah e 
abc 
4 - 
a" b" c" 
2a b c ’ 
the corresponding value of 
a + x 
fluent 
c+x 
-'b+z 
will be 
