XV111 APPENDIX THE SECOND. 



the old rule is nearer the truth than the new; since one 

 time with another, the renewal is made before the end 

 of the year in which the old life drops. This objection 

 must be valid to some extent ; and I proceed to inquire 

 how much weight must be allowed to it. 



Let a be the fraction of a year allowed for renewal : 



it is clear then that renewals (each accompanied by 



a fine) may take place in the year. Let the life A drop 

 at the expiration of the fraction 9 of a year, or between 

 and 6 + dd, the chance of which is dd itself, if A be 

 supposed equally likely to drop at any period of the year. 

 At 0-fci, then, the new life is put in; and if this new 

 life drop before 1 a of the year is gone, another fine 

 must be paid, and another renewal is made, which again 

 may drop before 1 a, and so on. But since the chance 

 of each additional renewal is very much smaller than 

 that of the preceding, it will be sufficient to take the 

 first only into consideration. Let it be supposed, then, 

 that not more than one renewal shall take place within 

 the year in which A drops. 



Let a be the chance that the life P drops in a year 

 after nomination, in which case we may call x a the 

 chance that it drops in any fraction x of a year. Then 

 d (i_0_ a ) a is the chance that the life A drops 

 between 9 and -f d 0, and that the next life drops 

 within the year, in which case another fine is to be paid 

 at the beginning of the next year. Consequently, neg- 

 lecting the interest of the fine in a fraction of a year, 

 the lessee has the chance d 9 (1 6 a) a of having a 

 second fine to pay, upon the contingency of A dropping 

 between 9 and 9 -f d 9. Integrate this expression from 

 6=0 to 9= l-o, and we have \a (l-) 2 for the 

 chance of a second fine : which, with the fine certain 

 upon the death of A, shows that 1 + \a (1 a) 2 is the 

 mathematical expectation of the fines to be paid, when 

 the probability of one renewal within the year is con- 

 templated, and another at the end, if necessary. 



A more complicated process, proceeding on the sup- 



