XXX APPENDIX THE FIFTH. 



both included), and also to joint lives, though not 

 with so much correctness, on account of the tables not 

 containing so many rates of interest. 



The following is an instance in which a deduction from 

 the calculus of differences will supply in a rough man- 

 ner the deficiencies of tables. There are none of these 

 for determining the mean duration of the joint existence 

 of two lives, but the defect may be supplied with suf- 

 ficient accuracy for many purposes, and particularly at 

 the middle and older ages, by the following RULE. 

 Let (3), (4), &c.' stand for the values of an annuity 

 on a single life, or on two joint lives, at 3, 4, &c. 

 per cent. : from twice (3) subtract (6) and reserve the 

 remainder : from (4) subtract (5), and having halved 

 the remainder, to it add the tenth part of (5), and 

 multiply the result by 9- Subtract the last product 

 from the reserved remainder, and multiply the differ- 

 ence by 10. The result increased by '5 in the case of 

 a single life, or by -25 in that of two joint lives, will 

 be something under the mean duration required. For 

 example, and to take a very unfavourable case, let the 

 Carlisle table be used, the life being 10 years old. 



(3) = 23-512 (4) = 19-585 



(5) = 16-669 



47-024 2) 2-916 



(6) = 14-448 



1 -458 



32-576 ^(5) 1-667 



28-125 



28-125 



45-01 the truth being 48-82 



