Survivorships. 255 
The general rule, therefore, in this cafe will become = D + 
E- 2 M. 
But when A is the oldeft life, recourfe muft be had, as 
in the fecond part of the preceding problem, to different fym- 
bols, and the value of the reverfion will then be found = 
S ab ms nt ~ S bs mt . nu 
— X 
ab 1 
S abc ' , ruse " 
ix -*+-£+5 + &c. + -X S-+2 + 3+ &c. + 
S abc' ruse ntc" « S bscf mtc nuc 
X + — 2 - 4 + &C. r x r r- + 
lube 
+ &C. 4“ 
3 ah ______ 
jix e J% + ?£ + ^ + &c. - , - 
3 abc r r z r 3 Z ahi r r ' 
2S rrtsc' . ntc" , ouc ' 11 
+ &C. + 
o W r/ v)t / 4- c" S mtc' nu . c -+ c" . * 
L. X — + .l-l-T— -I- &c. - — X — + + &C. + 
r' 1 abc v 
abc 
± + ?* ’ /+ - + &c. - 
«*£ >" 
S v wf . / _j . / + 
X ' I r 
r r- 
+ &C. *f 
± x ^ + £r^ + i^ + & c . + ^x^- l -i-^ 
tab r r* > J 
2 ab 
&C. 4 - 
c >. r ( r — _ c' a S « — S . C e J t . C C C 
+ + F + 
aac 
m -I x ? + ;4 + S + **. + .1 x = +£ + J +&c. The laft 
two, and the firft ten feries are=-2 M, the eleventh and 
twelfth feries are— .D, and the thirteenth and fourteenth feries 
=E; confequently the general rule becomes = D + E - 2 M, as 
before. 
Suppofing the lives were all equal, the above expreffio n wo uld 
be = 
s . V — CC 2.S 
x 737 . VTTCCC = l x r —L 
yZ'^CC — 2 LCC, which is known from other principles * to be 
the true value, and therefore the invefligation is right. As a 
further demonftration, however, it may not be improper to 
# See Phil. Tranf. Vol, LXXIX. 
©bferve* 
