PHILOSOPHICAL SOCIETY OF WASHINGTON. 115 



Then will 



Dx -f- Fa; = a constant quantity, 

 and therefore 

 AFa; = — aDo; = ; the symbol A denoting annual increment. 



But 6.Fx = F(x-\-l) — Fx = MJ)x-\-.05Fx 



Therefore, 



AF(a;+l) — aFx" = .OIaD.t + -OSaFo; 

 = — .0lAFa; + .05AFa; 

 = MaFx 



and AF(a;H-l) = aFo; (1.04) 



It follows that 



^F(x-\-n) = AFrc (1.04)« 

 and 



2 n 1 



F(x^n) — Fx = AFa; (l + 1.04 + 1.04 + 1.04 ) 



, Therefore Tm = -OHF(x+u)-Fx) + aFx 



aFx 

 To illustrate by a practical example : 



Required the period in years necessary to cancel in accordance 

 with the provisions of law, $750,000,000 of the public debt ; the 

 amount of debt outstanding, (less cash in Treasury) on the 1st of 

 January, 1880, being $2,011,798,504.87, and the amount of the 

 sinking fund accumulated up to that date $472,243,622 ; assuming 

 the future rate of interest which may be realized on investments to 

 he five (5) per cent, per annum. 



Bx = 2,011,798,504.87 

 Fx= 472,243,622.— 

 and AFa; = .01 Dx + .05 Fa; 

 = 20,117,985 

 + 23,612,182 

 ==. 43,730,067 

 and F(x+n') —Fx = 750,000,000 



Therefore 1.04 which equals 



. 04 (Fljx+n) — Fa;) + AFa; 73,780,067 

 AFa; ~ 43,780,067 



^^-reforen = '™-^ = m2m years. 



