estimation of the value of life contingencies. 
2 55 
ceived at the first of the equal periods^, after the time n — p 
that shall happen after the death of B ; provided he be sur- 
vived by A ; and that that event takes place between the 
periods n — p and m ; then if each separate period p, may 
throughout its duration, be considered within the limits of 
constant decrements, the value of that part due to the events 
happening between the periods nr and nr + P, (7 r being some 
one of the terms n — p, n, n+p, &c. ) from Article 1 of this 
section will be ' L °+*+iP r *+> = 
L a, b a, b 
x r 
ir+p 
J b+ir, a+n+Zp L a — \p ir+p ^a + v+^p, b+'rr+ p # 
L T . t x L * 7 * L 1 , , ’ 
- T * a a — 1 p, 0 
and 
\P> b —P 
if it be interpreted by n — p, n,n — 2/>, &c. the sum of the 
whole will be 
J a—lp, b-p 
L a,b 
a—\p, b—p 
J a—U P 
— — — . n 
a vn 
a + J pjb < 
If p be one year or unity, this will be 
J a— 1 > b 1 
^ a , b 
x (the an- 
a—i 
nuity for the time on the ages a — -" an( ^ ^ — *) — 
~~a 
annuity for the time on the ages a-fy and b. 
If the value of the contingency, due to the intervals between 
IT 1 IT i L XT 
2 7 t:a,b ' t ' 1 Tia-j-pib 2 ir.a,b-pp 2 ir-i-p:a,b 
L a,b 
it and nr + P> be written 
we may obtain the value in the forms of Messrs. Morgan, Baily 
and Milne ; for by reducing the form in the shape 
. r 
v+p 
L , _ L 
1 it\a,b v-\-p:a,b n+p 
2 l” “ • r 
J w : a-\-p, b 
” 2 L b L a,b-p 
interpreting 7r by n—p, n , n -\-p, &c. we have from Section 2, 
a, b 
L \a =t ^p h _^ ± b±p d 
2 L L , ’ 
a a—p, b 
