estimation of the value of If e contingencies. 283 
_n : a, b, c. Sec. , p — q n+p ^n+p : a. Sc, Sc c. ^ , / p + q Ji + p j 
• r ’ ^,ac. *•' 
p — q n + zp ^ n+zp: a , &, c, &c.\ / 
^ * L a,b,c,Se c. J ' 
/>— 77Z L 77t:g, fr, c, &c. \ , p_ 
+ - *p ' ' L a,b,c,&cc. J * 
'n+p : a, b. 
p+q m — p m — p:a, b,c. Se c. 
Z P ^ a , b, c, Sec. 
: a, b, c. Sec. ^>+f J 1 ^n: a, b, c. Sec 
• ~ j ' ^ K • • J ' ' ~ 
^ a y by Cy &C. P Uy by c, &c. 
+ 
J a, b, c, Sec. 
X ( r n +P . h n+ p : a,b,c,Sec. r ^ • ^n + zp : a, c, Sec. 
, m T \ m P — 1 L m:a,S,c, Sec. 
1 “ r • ^-‘m : a, S, c , Sec.) ^ * zp • L 
J a, c. Sec. 
rn -P + 1- L n:a,b,c,Sec.- rn -P-’l' L m:a,B,c,Scc. 
zp. L 
a,b,c, Sec . 
4 - P , 
1 n+p\ci, l, c. Sec., OF 
a,l,t • 
— ” TbTT~^ ( : a, b, c, Sec. r ” L m ; a> Cj & c .) "h » 
P ^ a,b,c,Scc. m 
As a particular example, if we take n — o, m infinite, p—\ 
r r 
q=.±, we have Jla, s, c . . f 3, c 3 &c. nearly, and there - 
0 1 1 1 
! £ i : a, 3, c 
J a, 3, c 
+ 
Cty by Cy Sir C# ) 
fore v fja, 5 ,c,&c. • i = nearly £ — £ r 
_ - 
i Lx ■ a S c Sec 
(or because r z £ - ■ * - ’ — : differs but little from unity) nearly 
J a, S, c,Sec 
equal to ^ T|a,^,c,&c. : that is an income of half a pound 
payable half yearly on the joint lives of the ages a, b, c, &c. 
is nearly equal to ■£ of a pound -f* the life annuity of one 
r 
pound on the same lives. If q — we have 
a, b, c, Sec. • 4. 1 8 
+ ~\a,b,c,sec. nearly; and f a *b,c, &c.. jr = f — jr 1 . 
a, by Cy &C. 
c, Sec. 
