250 
Mr. Morgan on 
third fines is = J x - ± x LI£z**l ; and that the 
fourth feries is = 5 x BC~^ABC - 1 x Thefe fe- 
O 3 . 
veral expreffioos being added together will be found = S into 
* . JTfk^afk BK-ABK 3 x FC - AFC + — X BC - ABC 
3 -, x 1 + 2 ^ 6 * 
m. PC-AFC^J^ BT- AB r ^ . FT - AP T 
6 br .$cr 2 ^ 
This theorem gives the exadt value when either B or C are 
the oldeft of the three lives; but when A is the oldeft:, it will 
be neceffary to exchange the fymbols a\ a'\ a'\ &c. for 
, S — U T-U, &c. and the fymbols m 9 n, o , &c. for 
^ZTc' c — r/~+ c \ c — c' + c' + c''\ &c. In this cafe the value 
of the given fum for the fir ft year will be found = — into 
abc _ bjc amt _ me _ M M _ *m± + miS for thg fecon( j year _ 
22 2266 3 3 
aba 
mic mtc nsc ntc msc" , mtc" nsc " ( ntc" msc' ^mtc 
into - -~ + T”T~TT + "6 T + ~< r + T" 
; for the third year = —3 into h — + — r- + 
“ o ^ 2 2 2 2 O 
2 2 
nuc' 1 ' ate"' . W" • c'+c" , . f' + f" 0 ' • c'+c" , ou . c' + c' 
j ~ + — r— ’ 
and fo on for the remaining years of A’s life. Hence the 
whole value of the given fum will be = -^x^ + ^ + ^- + &c. 
s ts . mt . HU , T~ s am ns ot & _ 
“5* x T + ?- + ?- + &C * + ^ X T + ?+,- + * c * 2«i * 
/ , jr 
ms nt ou $ $ abc ' . msc ntc q _ 5 * 
- + + &C ’ ~ Uic X ~ + ^ r+ ~ + 2 aber X 
?/uc' . nt . c' 4- c" 
’ I a 
c . S foe' wfc" ««f //7 « . s 
4 - &C. + 7 — - X 4 - 4 b &c. H 7 — x 
6 abc r > 2 H 2 obey 
mtc' 
