VALUE OF COPYHOLDS. XV11 



Simpson, &c., makes F infinite. Again, suppose the 

 life P certain to last one year and to drop in the second ; 

 1 "F 



in which case its value is , or ^ -, and F is a 



1 -\-r J&+ 1 



perpetuity of IL receivable at the expiration of every two 



E 2 E 



years, or - -. If A = P ^T" p tlle new formula 



becomes ; -, the old one becomes E, the value of 



a perpetuity of 17. receivable at the end of every year. 



I shall now show that the preceding rule agrees with 

 that of Mr. Milne; which is as follows: Let P be worth 

 an annuity certain of t years, and let v be the present 

 value of II. to be received a year hence. Then the pre- 

 sent value of all the fines, according to Mr. Milne, is 



A' + B'+C' + 



where A', B', C', &c. mean the present value of I/., to 

 be received at the end of the years in which the lives 

 severally drop. Since P is the value of an annuity certain 

 for t years, we have 



P = E-t><E, n-P=LT- ' = ?-<, 



r v 



E-A A' E-A 



and A = E77' whence i-r^l= TTP' 



from which the coincidence of the two rules is manifest. 

 The error of the old rule, by the Northampton Tables, 

 and at 4 per cent, (the best life being worth 17*25 years 

 purchase) is, that the result is 5-J- per cent, too great. 



The old rule, as Mr. Milne justly observes, is derived 

 from the supposition that the new life is put in at 

 the beginning of the year in which the old one drops, 

 instead of at the end ; which last was in the intention of 

 those who formed the rule. It may be said however, that 

 Y 2 



