237 

 it is evident that as — of the loan is brought up every year, 

 T=| • • • (1) 



P=| ... (2) 



a=pT ... (3) 



With regard to the first case : let a and p be as before, but let t be the 

 number of years it will take to pay off the loan by this method, and let v 

 equal 1 + the interest on one pound at the rate at which the Sinking Fund is 

 invested, so that if it is invested at 5 per cent, it will equal 1.05. 



Now at the end of the 



1 st year the fund will amount to p v 



2 nd ,-, „ „ (p + pv) v = pv + pv 2 



3 rd ,, „ „ (pv + pv 2 + p)v = pv + pv 2 +pv 3 



t th „ ,, „ pv + pv 2 +pv 3 + &c pv' 



but at the end of the last or t th year, the fund must equal a 



pv + pv 2 +pv 3 +&c pv 1 =a 



multiply by v and p v 2 + p v 3 + p v 4 + &c pv t + 1 =av 



subtracting pv t + 1 — pv = a v — a 



... a(v-l)=pv(v* -1). . (4) 



pv (yt -1) 

 ••• a = v _l • • • ( 5 ) 



P" ? (vt - 1) .... (6) 



When p is known the per centage required for forming a Sinking Fund 

 equal to p can be found by multiplying p by 100 and dividing by a. 



i a(v-l) 

 From (4) we get v* — 1= — — — 



v* = 



a (v — 1 ) + p v 

 PV 



... t log v=log < a (v— 1)+ pv > —log p v 



log J a (v-l)+pv ilogpv ,^ 



log V 



From (4) we also get p v t + 1 — (a + p) v + a=o . (8) 



From which v can be found by the following rule, known as Bernoulli's. 



1. Find by trial two numbers nearly equal to t. 



2. Substitute these assumed numbers for t and mark the error that arises 

 from each with + if too great, and — if too small. 



3. Multiply the difference of the assumed numbers by the least error, and 

 divide the product by the difference of the errors when they have like signs, 

 but by their sum when they have unlike. 



4. Add the quotient to the assumed number belonging to the least error 

 when that number is too little, or subtract if too great. 



