852 
DE. EAEE ON THE CONSTEIJCTION OF LIFE-TABLES. 
and the operation may be thus commenced. By adding the differences successively to 
each other and to '9969724, the successive values are found of Kja. 2 ^, 
kpso up to and including for males, where the series joins natui'ally the subsequent 
series, commencing at Xyjjg. 
-•000,0011,0490 
(constant) 
•000,0101,9910 
•000,0090,9420 
SK 
-•000,0872,7715 
-•000,0770,7805 
-•000,0679,8385 
_ 
1-996,9724,0000 
T-996,8951,2285 
r-996,8180,4480 
1-996,7500,6095 
In the actual operation the is subtracted from from and from ; it is 
therefore convenient to substitute for their present values the complements of and 
as thus all the series become additive. 
As Xko-\-t.2ho=^t-ki, and U 2 i+t.J> 2 i=t^k 2 , and generally it is evident that 
the is the^rs^ difference of the series 7d^-, and the whole series, 7d^, ffom to 
may be formed as in the subjoined example, where becomes becomes and 
so on. 
Healthy Districts. — Males, 
(constant) 
9-999,9988,9510 
Age. 
20 
sx 
0-000,0101,9910 
9-999,9127,2285 
S^=\p^. 
9-996,9724,0000 
4-584,1951,2769 
21 
0-000,0090,9420 
9-999,9229,2195 
9-996,8851,2285 
4-581,1675,2769 
22 
0-000,0079,8930 
9-999,9320,1615 
9-996,8080,4480 
4-578,0526,5054 
23 
9-996,7400,6095 
4-574,8606,9534 
24 
4-571,6007,5629 
J^ote . — The four last figures in the decimal portion of the series and in Xl^ may in 
practice be omitted. 
The corresponding values of Xp^ in the column headed Females, Table B, are inter- 
polated in the same way. And the Xpe^ Xp^^, Xps^, and Xpg^ are interpolated by the 
same methods, the series being continued backwards to Xp^^ and fomurds to Xp^oB ; the 
actual observations of age after the age of 90 furnishing results less reliable than those 
thus obtained, which bring a generation of 100,000 to theu last end in 107 years. 
The successive values of Xp^ in the period from the age of 3 to the age of 19 inclusive, 
are derived from Xp^, Xp^, Xp^^, and Xp^o, which represent Ug, u^, Ug, and As tlie 
terms of the series are here at unequal distances, the first differences cannot be derived 
from the preceding formulse. The S can in this and similar cases be derived ffoiii the 
proper equations by substituting figures for letters. But three literal equations supply 
formulae for finding the three first differences from any four terms of series of the Idiid 
which have been discussed: ^^o, which has a troublesome coefficient, can always be 
