estimation of the value of life contingencies. 
%39 
in question may be written in either of the two forms 0 
* L r X i~ T ( L r + n - L r + n + t) * ( L ^ + „+ \ + „ + *)>' or o 
J q + x 
J q+x 
• L r+ x — ( L r+»~ L r+»J1.J xL »+»+i*’ thatis by reduction in 
either of the two forms 0 
n 
^q + x + x 2 + n * + » H“ 
“ ^q + a ‘ ^r+n + t — • L r + n “f* ~ L r +n + f • + ^ or 
L^r+x • L r+ x L -f- L r+«+^ • L^+ni^ ; that is 
^q + x * ^r+x 
by using the abbreviation of Art. 4. Section 1, 0 
r Lr +«, J + m ”l“~Lq + n, r + n + i J"Lq+n + ^, r + n "{" “Lr + « + £,q + « + f 
x 
0 
or o 
n 
Lq-J-x • L r +x L r _j. Wj q + n+-|-£ -{- L r+w + fj q+« + -|£‘ 
Art. 2. Hence, if the deaths cannot be considered as the 
times, with sufficient accuracy, during the period x\ but dur- 
ing any portion of the limits 0 and s, s and s', s' and e", s" and 
s'", &c. the deaths produced in any portion of the same limits, 
are proportional to the portions of time in which they are 
produced, or may be considered so with sufficient accuracy, 
then as we shall have 0 
L q+x' L r + x ==0) We sha11 haVe 
Lq + x • L r + x — L r ,q~\~ ~L q, r+ £ - 
Lq + £ , rH™ ~T L ? 
T r + t >i+ t ; 
