XVI APPENDIX THE SECOND. 



value of the life, or the value of an annuity upon it ; r 

 the rate of interest per pound, and F the value of all the 

 fines. Then upon the new year's day next following 

 the extinction of that life, the whole value of the fines 

 will be 1 -f F, because the person who claims the fines 

 will have one pound to receive, and will then have 

 remaining the interest which we have called F. It fol- 

 lows therefore that F is the present value of I -f F to be 

 received at the end of the year in which the life drops : 

 or, by the well known formula 



where E is , the value of a perpetuity of 11. 



Secondly, Let the life already in possession be of the 

 value A, the lives at each renewal having the value P, 

 as before. Consequently, the present value of the fines 

 is that of 1 + F to be received at the end of the year in 

 which the present life drops : and this is 



.. 

 1+r ^ E + l 1+P 1 + P 



If an estate be held on any number of lives, with a 

 fine of I/, on renewing each, it is precisely the same 

 thing as if a similar number of estates were held on single 

 lives, and the present value of all the fines, the present 

 lives being worth A, B, C, &c., n in number, is 



l+P 



the common rule divides by P instead of 1 + P- 



A mathematical rule, when erroneous, is best exposed 

 in its extreme cases ; let us then suppose the life P 

 certain to drop in the year, that is, worth nothing. 

 There will consequently be a fine to pay every 1st of 

 January, and the present value of what I called F in 



the preceding investigation, is or E. The rule I have 

 given, shows that F=E when P=0 ; that of De Moivre, 



