estimation of the value of life contingencies. 241 
the interval of the times between our tabulated numbers of 
living, these may generally be considered to give the utmost 
limit of accuracy the tables will afford ; unless that there 
should appear sufficient regularity in the numbers of the 
tables to warrant a belief that an interpolation will offer a 
more accurate determination of the number of living for in- 
termediate intervals than the first differences only will give ; 
in which case our intervals s, s', & c. in, should, to obtain our 
confidence of their attaining the utmost accuracy, be taken 
smaller than the intervals of the tables, by taking their cor- 
responding numbers by interpolations. ' 
Art. 5. It may be observed with regard to Articles 1 and 
2, that when the object is only the calculations of the value of 
o 
o 
n + t 
hty -)- X * _j_ x 
L ? + x . L r _j_ x , the less developed form ° 
(L,. + „— L r+w + i ) . L ?+ „ + i, of its value, may be more conve 
1/ • 
nient than the form 0 L ? + x . L* + x — L r + n> q + „ + if -f- 
L r+n + Uq+n + W’ and that n being = £ + e'HH"+ &c. we shall 
have 0 
x, l-'r + x e ) * + P (^“V + s + 3 + e') 
^q + t-Yh' ~~~ + l + Lr + 6+ £ '+ £ ") • + s + 
Art. 6 . From Article 2 we have 
X 
0 
o 
«+*' 
— 0 
• -‘-'r+x £ 
2 ■+■ £ > r "t 1 + ^ 
-\-X 2 r B > *? + E 
T L <7+‘+* , r + .'+ J- L « + . + * , ,r + i + « ,; but ° 
q + X • _|_ x 
+ 
