VALUE OF COPYHOLDS. XIX 



position that any given number of renewals may take 

 place within the year in which A dies, gives the follow- 

 ing terms for the total probability of one or more re- 

 newals taking place before the end of the year : 



a, the chance that there shall be no renewal ; 



a(l-2a) 2 

 1 a - - - the chance of one only ; 



0-g) f _ *(l-3) the ehance of two onl 



and so on, the series being continued as long as the requisite 

 multiples of a are less than unity. Hence the chance that the 

 number of renewals shall not exceed n, is 



1- 



2.3.. n . n + 1 



which, in the case most against us, that is, supposing instanta- 

 neous renewals, or a = 0, is 



1 



Let the life P be one of seven years old, in which 

 case the Northampton table gives a =^$5= '0186. 

 Neglecting interest for the fractions of the year, and 

 remembering that the number of renewals is the num- 

 ber of fines, the mathematical expectation of ah 1 the 

 fines is the sum of 



(l-2a) 2 



a)3\ 

 /' 



2 v 2 2.3 



To this add a, the chance that the renewal fine certain 

 on the death of A, shall outrun the year, and we find 



a (12 cu 2 a 2 (l-3a) 3 



~T~ 2.3 



for the mathematical expectation of all the fines which 

 shall be paid in the year in which the present life drops, 

 including the chance of that life dropping too late to 

 renew it within the year, and of its being therefore re- 

 newed within a period not exceeding a of the year fol- 

 lowing. If a = 0, this becomes 

 Y 3 



