850 
DE. FAEE ON THE CONSTEUCTION OF LIFE-TAELES. 
preferable to any other. The logarithms oij)^ are required ; and to them it Tvill be con- 
venient to apply the interpolation directly. Any number of differences beyond four 
becomes cumbersome, and it will be therefore sufficient to give the general formula, 
which can be employed in deriving the first of either four or three orders of differences. 
Investigation of Formulae — Intervals egual. 
Let any numbers of a series be so related that the nih. from the first, Uq, is deter- 
mined by the equation (1.) — 
u. 
+ n[n—l) + n{n~l){n~2) + n(?i-l)(n-2)(re-3) 
I — Wq + 12 ^ 123 i 2 3 4 
. ( 1 .) 
and the first differences of the four orders, are unknown; they can all be 
determined from any five values of u,^. Now let 7i be successively la’, 2x, 3a’, 4a^; then 
the coefficients of u^^, u^^, u^^ can be found, to give the values of o\ and in 
four equations. But when x is ten or more the coefficients become large, and the nume- 
rical calculation laborious. It is therefore well to obtain the numerical values of 
in succession. Thus if the series is ascending or descending, the following 
are convenient forms. The upper rows of signs are used in the ascending, the lower 
rows in the descending series : — 
1 " Uix ~ 6 u2x ~ + u 
&4„+ - + - + “ , 
+ Mg* 3^2* 
+ 
— “T — 
i)S‘. . 
f - (^_i)y - 11) ^ ^ ^ 
+ V / _ 
- (^^- 3 ^ + 2 ) _ {7F-Qx^~ + \\x-%) 
6 °+ 24 °- 
( 3 .) 
(4.) 
(o.) 
It is necessary to be careful in deducing the successive values of ^ fi-om the values pre- 
ceding ; and before commencing their use their accuracy should be tested by inserting 
them in the checking equation, 
X may be any number. If only four terms are given, is assumed to be constant ; and 
being 0, all the terms into which it enters disappear. The- above formula, if this is 
borne in mind, are apphcable when or are assumed to be constant, and serve 
therefore to supply the differences when there are one, two, three, or four- orders by the 
most expeditious method. 
* It will be borne in mind that these imply first differences, or Jh/y, ^Uq. 
