XX APPENDIX THE SECOND. 



* a - 1 , c being 2 -7182818 

 a 



and when a = "0186 , = 1 *019 

 Hence '01 9 -r--0186 being 1-02, it appears that 

 1 02(1 -f- r) is an enormously exaggerated representation 

 of the fine which must be substituted for I/ in the am ended 

 rule to make the correction which might be suggested 

 by the advocates of the old rule. Allowing r= *035 , 

 it may be granted that the old rule is correct, in the 

 particular case before cited, if we suppose : 1. That no 

 time whatever is allowed for renewal ; 2. That the best 

 life which can be found is such, that 1 out of 59 of such 

 lives drop in a year, and ; 3. That the lessor is to receive 

 his fines at the beginning of the year succeeding that in 

 which A drops, with 3-Jper cent, interest, reckoned from 

 the beginning of the preceding year. To take a more 

 rational supposition, let us allow six months for re- 

 newal. Here a= \ and only the first term of the series 

 can be taken, which, with the interest, gives 1 -f %. 



It appears from the preceding, that the advantage of 

 the lessor over that which the rule gives him, is trivial, 

 except in this, that at the time when the rule supposes 

 him to receive one pound, he may have received and 

 improved one pound during a fraction of a year. There 

 is also another advantage which he has, and which neither 

 rule allows him. The renewal may have been made 

 before the time supposed in the rule, in which case the 

 existing life will be somewhat worse than that supposed. 

 To take all these circumstances into account, suppose 

 the life A to drop at the end of & of the year, in which 

 case the lessor is in possession of 1 -fr (0 4- (*)r at the 

 end of the year, and the lessee has a life * worth 



P 1 (6 -f- a) j A P, where A P is the decrement 



* Let it be remembered that the renewal may take place after the 

 beginning of the year, which is equivalent to having a better life than 

 that supposed. Algebra, as in other cases, strikes the balance of 

 positive and negative quantities^ without the necessity of introducing 

 several formula?.. 



