f 40 ] 


V. On the Method of determining , from the real Probabilities of 
Life, the Value of a contingent Reverfon in which Three 
Lives are involved in the Survlvorjhip. By Mr. William 
Morgan; communicated by the Rev, Richard Price, D. D. 
F. R. & 


Read January 29, 1789. 

I N a Paper which I had lately the honour of communicating 
to the Royal Society, refpedting the method of deter¬ 
mining the values of reverfions depending on furvivorffiips 
between two perfons from the real probabilities of life, I ob- 
ferved, that the inveftigation of thofe cafes in which three 
lives were involved in the furvivorffiip (though attended with 
much more difficulty) might, however, be effected in a fimilar 
manner. The further purfuit of this fubjefl has now con¬ 
vinced me that, as it is never fafe, fo likewife it can never be ne- 
ceffary to have recourfe to the expectations of life in any cafe; 
and that the folution even of thofe problems which include three 
lives is far from being fo formidable as at firft fight it appears to 
be. I am fenfible of the impropriety of entering minutely in 
this place into the vaft variety of proportions which refer to the 
different orders of furvivorfhip between three lives; but as the 
following problem feems to be of conliderable importance on 
account of its being applied to the folution of many other pro¬ 
blems, the demonftration of it, perhaps, may not be thought 
an improper addition to my former Paper. 

PROBLEM. 


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Mr. Morgan on Surmvorjhlps . 


41 

P R O B L E M. 

Suppofing the ages of A, B, and C, to be given ; to deter¬ 
mine, from any table of obfervations, the value of the fum 
S payable on the contingency of C’s furviving B, provided the 
life of A fhall be then extindt. 

SOLUTION. 

Let a reprefent the number of perfons living in the table at the 
age of A. Let a', a", a / / , a"\ &c. reprefent the decrements of 
life at the end of the ift, 2d, 3d, 4th, &c. years from the age of 
A. Let b reprefent the number of perfons living at the age 
of B, and m, n, 0, p, &c. the number of perfons living at the 
end of the ift, 2d, 3d, 4th, &c. years from the age of B. In 
like manner let c reprefent the number of perfons living at the 
age of C, and </, e,f, g, &c. the number of perfons living at the 
end of the ift, 2d, 3d, 4th, &c. years from the age of C. 
Let r alfo denote the value of jT. 1 increafed by its intereft for a 
year. In order to receive the fum S in the firft year, it is 
neceflary either that all the three lives fhall have died in that 
year, A having died firfl, B next, and C laft ; or that only the 
two lives A and B fliall have died (A having died firft), and 
that C fhall have lived to the end of that year. The proba¬ 
bility that the three lives fhall die in the firft year is 

—■ The probability that they fhall die in the order 

above mentioned is —-. The probability that both 

A and B fhall die in the fir ft year is —Half this frac- 
Vox.. LXXIX. G 


tion. 



42 


Mr. Morgan on Survivorships. 


tion, or —~ b is the probability that the death of A fhall 

happen before the death of B in this year. The probability 
that C. fhall furvive A and B, retrained to the contingency of 


As having died firft, is 


> — m 


2abc 


The value therefore of the 


film S for the firft year is S x 


a' • b — m . c — d 
6 . abcr 


> b—?n . d 
2 cibcr 


abcr 


a'be a'me a'db a nid 

~b ~’~V 3 ~ 3 




. In the fecond year the pay¬ 
ment of the given fum will depend on either of four events 
happening. Fir ft, on the contingency of all the three lives 
dying in that year, A having died firft, B next, and C laft. 
adly, On the contingency of B’s dying in that year, C’s living 
to the end of it, and A’s dying in the firft year, ^dly, On 
the contingency of B’s dying after A in the fecond year (both 
of them having furvived the firft year) and of C’s living to the 
end of that year. 4thly, On the contingency of A’s dying in 
the firft year, and of B and C’s both dying in the fecond year, 
B having died firft. The probability of the firft contingency 

The probability of 


is exprefted by the fraction 
the fecond by the fraction 
third by the fraction — ' — , 

J 2 . abc 


a" • m—n . d—e 


t> . abc 


a . m — n . e 


abc 


The probability of the 


m — n . e 


fourth contingency by the fraction 


And the probability of the 
Thefe feve- 


a '. m—n . d—e 


labc 


xal fra&ions, therefore, multiplied into will be the value of 
the given fum for the fecond year, and may be eafily found = 




S a"dm a"dn a" cm a”en . a'cm a'en , a!dm a f drt - ... 

like 


6 

I 


3 — .3 





Mr. Morgan on Survfootjhipt. 4j 

manner the payment of the given fum in the third’year will 
depend on the contingency of the fame number of events 
as in the fecond year; that is, it will, firft, depend on the 
contingency of all the three lives dying in that year, A having 
died firft, B next, and C laft; 2dly, on the contingency of 
B’s dying in that particular year, C’s living to the end of it, 
and A’s dying in the firft or fecond years; 3dly, on the con¬ 
tingency of B’s dying after A in the third year (both of them 
having furvived the two' preceding years), and of C’s living 
to the end of that year; and, 4thly, on the contingency of 
A’s dying in the firft or fecond year, and of B and C’s both 
dying in the third year, C having died laft. Thefe feve- 
ral contingencies are exprefled by the refpe&ive fractions 


€1 


n — o . e—f 


6 • abc 


a' 4 -a" * n~o\f 
abc 


a ! f . n — o . f 
2 abc 


a ■\ -a n . n—o • e — f 
%abc 

the third year will be = ~ 


Confequently the value of the fum S for 


a!" . eo a"' .fn a'" . fo 


+ 1 


.fn a' + a" .fo t a'+ a" • en a + a" • eo 
2 ~~ 2 ^ 2 2 
in the fame manner the value of the fum S for the fourth 


And by reafoning 


year may be found 


' abcr 4 6 ~~ 6 3 3 


.a' + a^+a'". go a'+ a"+ a" . g/> t a' + a" + a'" • °f a ' ^ a " + a "'* fP 
v ~ - - — 2 + 2 ~ 2 

If either B or C be the oldeft of the three lives, thefe 

feries continued to the extremity of that life will exprefs 

s 

the whole value of the reverfion, which will be = — x 


a bc a"dm ^ a ,tr en 

abcr a for 2, abgr 1 




abcr 


akr 1 


44 


Mr. Morgan on Survkorjhips. 


/ , // , /// 
a -f 


„ S- ci'nd af'oe . a ! nf . />/’ 

" + &c - - 6 * w r + *i;r* + s?+-*;.■+ &<=. - 


S a dn a -j- a ''. *0 a! a" a!fp « S a 'dfr a " , 

X—+-TTZT- +-+ &C - +rx—+—r + 


2/' abcr 


abcr~ 


abcr abcr 2 


tf" 7 ./* a 7777 -** , , S r a"-fra". fn , 

- +&C+-X _ + ___+--p- 


3 abcr 4 

S 


2r 


abcr 


a'md c^en a.Jo a"”. gp 

X abcr abcr 2 abcr 3 


+ &C. 


+ 

s 


■4" &c* 


2r ~ 


. g/> 


abcr 3 


+ &C. 


In order to fum up the firfl and fecond of thefe ferles let (S 
reprefent the number of perfons living at the age of F, a 
perfon one year younger than B, and jc the number of perfons 
living at the age of K, a perfon one year younger than C. 
Let FK, BC, AfK, and ABC, reprefent the value of an 
annuity on the two and three joint lives of F and K, of B and 
C, of A, F and K, and of A, B and C refpedtively; then 

S . 0 

6 


, . r . S a'be a''dm 

will the fenes zX — + — + 


abcr 3 


+ &c. be 


6 . be 


X 


be a —a' .be dm 
far afar (3 a r s '" 


a—a'—a" * dm a'dm 


a-a 


afar 2 


a' ~Y a" . en , S * £* IK-AFK 

. „———— 5 e£C. — —— X > —• 

a$*r^ be 6 


a fix.? 2 far 3 
S 


afar 3 


(- 


x a 

6 r abcr 


a' + a' ‘ 


en 


abcr 2 


&c. 


S dm 
r X 


a— a '» dm en a —a'—a' 
“T” 7 —— 1 


en 


br ber abcr * her 2 abcr % r J 6 r * 


BC - ABC. The fum, therefore, of the two firft feries, or of 


S a'be a ". dm 0 S a'dm 

x : + ”T~:r“ + &c. + “X 


a +■a 


abcr 


abcr 2 

S 

3 ' 


2.r 


abcr 


abed 


en S , 

— + &c. IS = ^ x j 


k T x ^ “ ABC. Again, let P reprefent a life one 

year older than B, and let BK, PC, ABK, and APC, repre¬ 
fent the values of annuities on the two and three joint lives of 

B and 




Mr. Morgan on Survivorjhips. 45 

B and K, P and C, A, Band K and of A, P and C: then 

s 

the fum of the third and fourth ferles, or of - C x 


a! me a'nd 


aber * aber 2 


+ &C* 


S a'dn a' -f a n , eo n . . 

X” + “^r- + &c. being 


2 r A aber 


S. « 

be 


me a — a f .me dn 

+ 


xbr 


ab*r 


b*r 


a —a!—a" * dn eo 
b^r 3 


ab* 


abw 1 


- , &c. 


(— 

\ v 


S a'dn a' + a /f . eo 


will be = 


X + _/. ..2 9 &C. -— ) 

S ■ X . BK-ABK S rn . BC - AFC 


3r " aber 


aber 2 


S . 77i a ; dn a' -r a" . eo Q 
- X -+ ■-— + &Ce 5 


3 ^ 


acmr 


3 r 


The fifth fie- 


* S a f db a" em 0 , S ft db 

rieS ' - X ^ ^ + &C - 13 = “ * T * 


( 3 cr 


a —a / . db ; *772 


a~-a'—a"*em fn a —a'— a" — a"'.fn e S a f em a' + a^./n 

■ - - -- JL r — - —■— - ? &C. — — X —f -r 7—5—“,& C * 

after * ftcr^ after 3 ? Y aber aber 2 

= (putting FC and AFC for the values of the two and three 
joint lives of F and C and of A, F, and C)~ x - ‘ Fc ~ AFC 


b 


.1 x tlH 1 t.. + a ....d- &g. The fum, therefore, of the fifth and 

Y aber * aber 


t r . r S a 1 db a"em Q S a*em a f -\-a" . fn 

fixth feries, or of - x ^ ^ &C -+Tr x 7 ^ + + &c. 


is — — x 
3 


aber ‘ aber 
(S . FC —AFC /S 


ahr 2 


+ ( 


6 r X otar * 


a r +a" . fn 
abrc z 


, &C. — ) 


Sxi 

6 tr 


X 


£•??,. a — a 1 . era fa a <3 a . fn - t r i t*/' 

-—4. — _ --j- P&c. lr I denote a life, one 

bdr abdr ■ Wr 

year older than C, and BT, and ABT denote the values of the- 
two and three joint lives of B and T and of A, B, and T, this 

6 r x - ~c -» and conlequently the fum 


lafl feries will be; 


of the fifth and fixth feries will be = — x - '■ - C - --~£-j- x 

3 b or 


d. BT—ABT 


Laftly,. the feventh and eighth feries, or ■ 





Mr. MoftG &¥r on SurvhorJfjtpi. 


46 


S a 1 md a"en 0 
- X — - + — 2 + &C. 


abcr aba * 


S a en a a n . fo n 

— x ~r~ + ——— + &c. are 

2r tf/'cr <3^0* 


X' 


dm 

her 


a — <1 • dm en 
- ^ ■ 


a — a — a 


abcr 


bo 4 


abcr' 


; + &c. 


tx * abcr 


a'en a! -{- a'\fo 

-i+.& c . 


abcr 


x BC ~~ ABC 


a 


S md en 
* T * ~r 

be mdr 


a — a'.en • of 
amdr mar 2. 


N .of 


■&c. 


) ~ x — - - y - —— , where FT and APT 

amdr J 0r be 

rep refen t the values of the two and three joint lives of P and ; 
T, and of A, P, and T. If thefe feveral expreffions be added toge¬ 


ther, &c. we fhall at laft have 


S.X <?. FK-AFK 


6 c 


BK. - ABK 


x FC - AFC- 


3 * 


y 


x BC - ABC 


S . m 
3 br 


xPOAPC-t- 


s. d 


x BT ABT - m - 1 1 ~- for the value of the fum 

6 cr o 

when either B or C are the oldeft of the three lives. 

In order to determine the value of the reverfion when the life 
of A is the oldeft of the three lives, let s, /, u, w, &c.be the num¬ 
ber of perfons living at the end of the ift, 2d, 3d,4th, &c. years 
from the age of A, and let b\ b'\ b"* i b'"-, &c. be the decre¬ 
ments of life at the end of 1, 2, 3, 4, See. years from the age of 
B; then, by reafoning as above, the value of the fum S for the 

iirft year will be exprefted by the feries ~~~ ~ "hi 
sxb . a ~ s . ^ £ dJ . t j ie f econc i y e ar by the feries ~ +; 


2 abcr 
S • b ,f . s—t . e 
< Zabcr z 


S • b tf a-~s . e S ♦ b f ‘ 

■ -r —— 


abcr 


%abcr 2 


e , for the third year 


. , r . S . V" .t-tt. T^r , S . V" .t-u.f S . V" . a — f . f ■ 

by the feries- tabO~~ *—~ +1 


zaber 3 


abcr 3 


S . b nt . a-t . 


ilaber 3 


—, and fo on for the remaining years of A’s life. 


Thefe feveral feries may be found • 


aVc b'cs ab f d b f ds 


abcr 


X 


6 


+ 







Mr. Morgan on Survlvorfhlps. 


a’o'd at/c 

‘—- + — + 
2 r 2 * 


s 

abcr 


sb"d 


X —* 


b"dt 

~6~ 


sb"e bd't ab"e ao"d 


~~T~Y + 


' +' 


+ 7; 


£<.r 3 


X 




b" r eu 

' ~6 - ' 


' 6 ’ 


. —- 4- 4. 1 , &c. &c. Let a reprefent 

3 2 2 


the number of perfons living at the age of H, a perfon 

one year younger than A; let N denote a perfon one year 

older than A, and let the feveral combinations BN, BNC, 

AB, &cc. denote, as in the former cafe, the values of annuities 

on the joint lives of B and N, of B, N and C, of A and B, 

&c.; then by proceeding in the fame manner as in the fore* 

aVe sb"d tb"'e 

: + + &c. may 


going demonflration the feries 

be found 

Vea 


3 ac 


X HK - HBK. - 


3 abcr ~ r 3 abcr^ 

AC — ABC 

._ * 

9 


' baler* + &C- _ be X ' 


3 r 


6 ar 


'^abefi 

y r . Vcs b"dt 
the leries —r ~ 4- 

o abcr baba 


x NC — NbC; the feries 


“ x HC - HBC —d- x AT - ABT; 


ba 


a IL , S-+&C = 

baber "** baber z "* baber* 

. . r . Vis Vet Vfu 

and the feries —+— + — + &c. 

3 MT — NTB. Thefe four feries, therefore, fuppoling them all 


6cr 
AC-ABC 


sd 

$acr 


to be pofitive quantities are = 2 x 


* . HK - HBK . AK - ABK 


~ x HC - HBC + — x AC - ABC 

ba 2? 


7— x i\C- NBC — x 

bar jer 


AT —ABT s . NT —NBT 


r . b’d b"e 

ieries 

2 ber r 


b"'f 


With refpedt to the two remaining 

b’c b"d . i'"- 


f -f &£*. • . and 


'zber^ 2 bo* + ^ C,J 


-Xhr T Zlcr 2 - * — -- 2 ber 

thefe, it is evident, are to he continued after the deceafe of A 
till the extindtkm of the joint lives of B and C, and have 
been already proved in the folution of the fecond problem 
in my former Paper, to denote the value of the given fum 
payable if C flhouid furvive B» Let this value be repre > 



48 Mr . Morgan on Survhdrjbips, 

fen ted by R and the fum of the foregoing expreffions (or 


a . HK—HBK AK-ABK 


3 C 


X 


+ 6 x HC — HuC, &c.) by M, then 
will the value of the fum S (when A is the oldeft of the three 
livesl be = S x R — M. Q. E. D. 

If the three lives be equal, the value of the given fum for 

i r n 'lit ^ S. 7 — 4 2 • d 

the firit year will be = - 4- 


2 . r 


c i 2a 3 3 dd 

Sx -+ 6 t 5 ,-~ ! 


the value of the' fame fum for the fecond year will be ^ 
S . ~d —d 3 


> bc 3 r 2 


+ 


S .d"-eV 
TV“ 


c , S . d—e .c-d.e , d—e 
*■ *T “13 "r 








= s 


x 


6<A 2 6f 2 r 2 


2d 3 

bc^r 1 


■.u- ye _ the value for the third year will be: 


S x 




3 ff 




ye 


and fo on for the other years to 


6cV 3 6t r 3 6cV 6<V* 

the extremity of life. Let CC and CCC denote the values of 
the two equal and three equal joint lives, the fum of thefe 


feries may then be found ■ 


•— x — + 2CCC — 3CC 4 

or •* ' 


3.CC-2CCC 


= (fuppofing the perpetuity, or to be denoted by V) 

X L 3 X V -2.CC- 2CCC. 

or 

It muff be here remembered, that from other principles it is 
well known, that the number of years purchafe expreffing the 
value of an ejlate ox perpetual annuity to be entered upon at the 
failure of two out of any three equal lives is, “ the difference 
“ between three times the values of two equal joint lives, and 
“ twice the values of three equal joint lives fubtrafted from 
44 the perpetuity,” or V — 3CC — 2CCC. The value, there¬ 
fore, of fuch a reverfion, fuppofing it to depend on the failure 
of the three equal lives in any one particular order, is (fince 

there 


2 




49 


Mr. Morgan on Survivor/hips, 

there are fix fuch orders equally probable) 4 x V — 3CC - 2CCC. 
But it appears, from the corre&ion explained in Dr. Price’s 
Treatife on Reverfionary Payments, Vol. I. p. 34. that the 
value of a reverfionary fum is always lefs than the value 
of an equivalent reverfionary eft ate in the proportion of 
1 to r. The fum being S the equivalent eftate or perpetual 

annuity is always Sxr—i; and confequently the value of 
the fum S depending on the ceafing of three equal lives in any 
one particular order and thus determined, is the fame with 
that determined by the foregoing invefligation, that is, 

— x !- x V - 3CC — 2LC, The invefligation, therefore, is 
right, and the correction and invefligation demonftrate one 
another. 

But the foregoing expreflion for determining the value of 
the reverfion in this particular cafe is not only obtained imme ¬ 
diately from the feries, but alfo from the two different rules 
which have been given for determining the value when the 
lives are unequal; and hence a proof arifes of the truth of 
thefe rules, as well as of the reafoning upon which they arc 
founded. Thus the firft rule, fuppofing the lives all equal, 

, ** KK-CKK dd TT-CTT ~r -7 CC-CCC 

becomes x-- - — x -5--7- *- - + 

* CK-CCK d CT-CCT . . r . , , 

— x-7-x —-7— , and the lecond rule becomes 

c 6 cr 6 7 


V - CC . »- T x 2 ' KK-CKK, dd TT-CTT r-\ CC-CCC 

--- r X - d-- X —--X -- 

2 r c 3 cc , r 3 r 3 

* CK-CCK d CT-CCT T , , 

x _--f- — x - • Let the value according to 

c 3 c> 3 ° 

the firft rule be denoted by L, and the lecond rule will be = 


1.v-cc 


t — 1 . cc -ccc 


-( ■= IA Hence 3I 



FI 


r — { 




5° 


Mr. Morgan on Survlvprjhips . 

~ G lr. 2 • r ~~* • CC ~S££ andL = r -=l x V - 3CCT^CCU. 

2r 6 r J 

Q. E. D. 

Were vve pofTefled of complete tables of the values of annui¬ 
ties on two and three joint lives, the preceding rules would give 
an ealy and exafl folution of this problem in all cafes. But 
as l'uch tables, computed for every age, would be a work of 
immenfe difficulty, efpecially in regard to the values of three 
joint lives, Mr. Simpson’s rule for approximating to thefe 
from the given values of the two joint lives, has hitherto been 
adopted, and it feems upon the whole to anfwer the purpofe 
very well. In the prefent problem it is attended with no other 
inconvenience than increafing the labour of the computations 
for the values of the reverlions derived from it appear in gene¬ 
ral to be perfectly correct. This is more fully afcertained by 
a table which Dr. Price has given in his Treatife on Rever- 
fionary Payments (Vol. II. Table 37.), of the values of three 
equal joint lives computed at 4 per cent, from the probabilities 
of life at Northampton. By the affiftance of this table, 
when the lives are of the fame age, it is evident, from what 
has been already obferved, that the exafl value of the reverlion 
may be eafily obtained. The few following fpecimens com¬ 
puted from it, and compared with the values of the reverlions 
deduced from the firft and feeond of the preceding rules, de- 
monftrate the accuracy of thofe rules: for, notwithftanding, 
the approximated values of the three joints lives have been ufed 
in every inftance in which the rules have been employed, yet 
the refults approach fo near the truth, even in the laft Rages 
of life, when the decrements are moft irregular, that, though 
derived from thefe approximations, there can be little doubt of 
their corredtnefs in almoft every other period of life. 

6 Common 



Mr. Morgan on Survivor/hips. 51 


Commonage. Exa& value** of £. too. Value of £. too. com- Value of £. 100. computed 
computed from Dr. Prick’s puted from the fir ft of the from the fecond of the fore- 
Tables of the values of two. foregoing rules, and from going rules, and from Mr. 
and three equal joint lives. Mr. Simpson’s approxi- Simpson’s approximation to 
mation to the values of the values of three joint 
three joint lives. lives. 


JO 

12,000 

12.005 

- 12.000 

75 

12.944 

12.943 

12.943 

80 

13.840 

O 

fcH 

OO 

cr> 

- 13.880 

85 

34.4504 - 

14.670 

14.340 

Mr. 

Dodson and Mr. 

Simpson §, 

are the only writers 


who have folved* or rather who have approximated to the 
folution of this problem. But the former* by deducing his 
rules immediately from a wrong hypothefis* having rendered 

** That is, one-ftxth yzn of the whole reverfion. 

+ The feveral reverlions in this column, when computed from Simpson’s 
approximation to the values of the three joint lives, are 12.012, *2.933, 13*84/* 
and 14.803 refpedfively; which upon the whole differing nearly as much from 
the real values as thofe in the two other columns afford a convincing proof* 
that the very fmall deviation from the truth in thefe latter values proceeds not 
from any inaccuracy in the rules themfelves, but folely from having ufed the 
approximated inftead of the real values of the three joint lives. And this alfo 
will account for the difference in the values by the firft and fecond rules. Were 
thofe values computed from tables which give the correct values of two and three 
joint lives at all ages, they would come out exactly the fame. In the two firff 
examples, where the values by one rule are true* it appears, that the values hy 
the other rule are equally fo. In the two lafl examples, where the values are not 
quite fo accurate, it may be obferved, that they differ as much in excefs by one 
rule as they do in defeat by the other; which mull in general be the cafe from 
the very nature of thofe rules; for if L (or the value by the firil rule) be 

greater than the truth, the difference between r - --- anc ^ ^ ( or 

Value by the fecond rule) muff be lefs than the truth; and* on the contrary, if 
L be lefs, this difference will be greater than the truth. 

% See Dodson’s Mathematical Repofitory, Vol. I 1 I« Qucftions 42* 43* & Ge 

§ See Simpson’s Select Exercifes, Prob. 38. 

H 2 


moft 



t;2 Mr. Morgan on Sur'uivorjhips. 

moft of them (efpecially thofe in which three lives are con¬ 
cerned) of no ufe, it will be /unnecefl'ary to take notice of 
what he has done on the fubjeft. With regard to the latter, 
whofe rule is not only the foie guide for determining the value 
of this reverfion., but alfo the fource from which a great va¬ 
riety of other problems are folved, perhaps it may not be 
improper to examine how far his folution is to be depended 
upon ; and the following examples have therefore been com¬ 
puted for this purpofe. 


TABLE I, 





Value, by Simpson’s rule, of 

True value of the fame 


Ages of 


IOO. payable on the contingent 
reverfion fpecified in this pro¬ 
blem, when either C or B are el- 

reverfion computed front 
the firft rule in the fore-* 
going folution* 



deft, according to the Northamp¬ 

c. 

B. 

A. 

ton Table and at 4 per cent . 


8 o 

7° 

40 

1.926 

Os 

75 

65 

25 

1.873 

I.O32 

65 

50 

15 

2.090 

I.69O 

70 

80 

40 

- 6.615 

6.II7 

5° 

65 

*5 

5.580 

3- § 79 

GO 

78 

20 

2.583 

1.982 

45 

60 

I 2 

> 

in 

t-n 

> 

8 

4-133 

60 

45 

12 

2 . 2^2 

1.686 




TABLE IL 





Value of the fame reverfion by 

True value of the fame 


Ages* 


Simpson’s rule, when A is the 

reverfion by the fecond rule 

c. 

B. 

A. 

oldell of the three lives. 

in the foregoing folution. 

24 

6 5 

75 

34.636 

31.792 

65 

24 

75 

6.305 

7*895 

49 

9 

69 

7-35 1 

5.96° 

18 

78 

78 

" 37*554 

33 ,OI 9 


TABLE 



Mr. Morgan on Survivorfhips. 


53 



TABLE III. 



Value of the fame reverlion by Simp¬ 

True value oi 

Common Age, 

son’s rule, when the ages of the 

fame reverfion. 


three lives are equal. 


70 

I3.2O 

I 2.00 

75 

- - 14.98 

- I2.94 

80 

16.58 

13.84 

8 5 

17.86 

14-45 


By comparing the values in the preceding tables, Mr. Simp¬ 
son’s rule appears in almoft every inftance to be exceedingly 
incorreft. Even when the lives are equal (in which cafe it 
might have been expected to be fufficiently accurate) it feems 
to deviate, in old age at lead, fo widely from the truth as to 
be unfit for ufe. When C or B are eldeft (which, however, 
is a cafe that does not often occur), the refults fometimes 
exceed the truth one-half.\ and generally by more than one- 
third of the real value. When A is the oldeft of the three 
lives (which is the mod common cafe) thefe refults are erro¬ 
neous in nearly an equal degree. Nay, in fome cafes, Mr, 
Sampson’s rule is not only wrong but abfurd. Thus, in the 
laft example in the fecond table, the value of £. ioo. payable 
on the contingency of C aged 18 furviving B aged 78 
after A aged 78, is by this rule rr f. 37.554. The value, 
therefore, of the fame fum on the contingency of G’s furviv¬ 
ing A after B is alfo 37.554. Hence the value of f. 100. 
on the contingency of C’s furviving A and B (without the 
reftriftion of one dying before the other) is 2 x 37.554 = 
£. 75.108*. By another rule of Mr. Simpson +, the value 
* See Simpson’s SeltExercifes 3 . Pxob, 39, f Ibid, Prob* 32* 



54 . Mr. Morgan on Survhorjhips. 

of £. ioo, on the contingency of C’s furviving B only, is no 
more than £. 74*. Now it is felf-evident, that this latter 
value, inftead of being lejs, ought to have been greater than 
the former, inafmuch as the probability of C’s furviving only 
one life muft be greater than that of his furviving two lives. 

Many additional inftances might be produced in which this 
rule, being made the bafis upon which the folutions of other 
problems are founded, leads to conclufions equally erroneous. 
But thefe enquiries would be improper here; and I fliall 
only obferve, that had the foregoing examples been computed 
from the Sweden or London, inftead of the Northamp¬ 
ton Table, this rule would have appeared to be ftill more in¬ 
correct than it does from thofe computations. 

When Mr. Simpson wrote his SeleCt Exercifes, he was in a 
great meafure obliged to have recourfe to De Moivre’s hypo- 
thehs, for want of thofe excellent tables of the real probabi¬ 
lities of life, and alfo of the values of fmgle and joint lives 
which have been fince publiffied. Had he been poflefled of 
thefe, it is moft likely that his fuperior abilities would have 
directed him to a more accurate method of inveftigation. At 
prefent there can be no juft reafon for ever recurring to this 
wretched hypothecs. The folutions of all cafes of two and 
even of three lives may be effected without much difficulty 
from principles ftriCtly true. But I muft here take my leave 
of this fubjeCt, hoping that its importance may engage other 
mathematicians to the further profecution of it. 

* The true values are £. 66.038. and £. 74,884. refpe&ively. 





