RULE FOB SURVIVORSHIP. XX111 



of the rule,, <j>ti& assumed = a + A a . and i//=/3 + A/3.Z, 

 t being measured from the beginning of the (n + l^th 

 year. Hence 



f$t. ^'t.dt (from #=0 to t=l) = a A/3 | AaA# 



which is the common rule in a different form. 



Let us now suppose (t being measured from the be- 

 ginning of the (n + 1 ) th year) 



* 1 



<$>t = a, +Aa.t + A 2 a. - 



where the differences constitute series of rapidly di- 

 minishing terms. The only term of the second order 

 which this addition to the hypothesis introduces into 

 -/< ^'t.dtis aA 2 /3/(^-^) dt which is =0, when 

 taken from t=0 to =1. Consequently the errors of 

 the rule are all of the third order. 



To give a notion of the amount of error, extend the 

 preceding formulae to terms of the third order, and form 

 the integral, reserving only the terms of the third order. 

 The final result is as follows : If x and y be the num- 

 ber of the living at the age of A and B, and if , b, c, . . . 

 be the numbers alive, at n } n + 1? years older than A, 

 and p, q, r, . . . at n, n -f 1, ... years older than B, then 

 the probability that A shall begin to survive B in the 

 course of the (n + 1) th year of the calculation, is 





2 xy 



the first term being that generally used, and the second 

 a correction which ought always to be applied in those 

 parts of the table in which the yearly decrements are 

 not equal. 



The demonstration of the preceding will be easily 

 arrived at by the indication which I have given, by any 

 one acquainted with the integral calculus. To those 

 who have not that advantage, reason may be shown in 

 the result, though not for the result. The preceding 



