Mf . Morgan on Survivorships. 
*59 
and by the fourth rule it becomes = S into -~ I • V ~~ L 4. JL 
^ 6r I if. X 
2CK — 
CCK 
. CKK 
bcc 
d d 
~~-pr X2CT 
CCT 
3 C 
dd 
6 ccr 
"f* T7-T x 
1 + C XT. If the values of the joint lives in each of those 
rules be resolved into their respective series, all the expres- 
sions, except the first, will be found to destroy each other, and 
the general rule in all of them will become simply— s, ~ •^- L 
which from self-evident principles in this particular case, is 
known to be the true value. A similar result may likewise 
be immediately obtained from the series themselves ; for the 
value of S for the first year is easily found in this case to he =,- 
C “ 7 — 
® 1 d . 
r X ~ ~77 + 
dd 
2 cc 
+ — — • — 4- 
1 2r. 2 c • 
dd d 3 r _ c — 
17 ior the second year = ~ x ee 
2 cc 
ee 
2 cc 
~r, for the third year = ~ x 
J Z- 3 2 CC 
4. _£_ f_ f g 3 7 p~ 
2 c 2 c ' 2C 3 ■ and so on for the other years. 
Hence the whole value is = 
S. r— 1 . V_l 
6 r • 
S- E- D. 
It is to be observed, that the fractions h ~ m - d -^g b-n.TTj 
o j 2 abcr z 9 2 abcT 3 9 
&c. do not accurately express the value of S on the second 
contingency in this problem ; but that according to the lemma 
b —n . d— e . a b ■ 
zabcr 2 
e-f 
&c. 
they should have been 
In order to determine how near the former Approach to The 
* When B is the oldest these fractions are __ 
When A is the oldest they are 
BT-ABT-T^A'f 
C'— VC'_ BC + AB C d 
* r + ler X 
C'_B'C'_ AC + ABC 
— + 
2 r 
L 1 2 
