454 VALUATION, DEPRECIATION AND THE RATE-BASE 
The basic formula for $100 derived from (28) from which all 
other formule are derived will be 
100 7 
n= (1 +0)" ogy (35) 
Note. — This formula is for the special case in which the 
amount of the annuity is $100 which fact should not be over- 
looked in applying it. 
The accrued amortization in m years will be: 
(1 +2) — ‘] 
An = re ————T Al 
roo] Gry: (36) 
The remaining investment at the end of the mth year will be 
100 — Am = 100 |: i naan . 37) 
The current amortization a, in the mth year will be 
0 i (1 + om! | 
Onl = wate al (38) 
For depreciation, as ordinarily though erroneously * deter- 
mined, these formule may be written for an original value of $100: 
100 7 
aq, = Gi —1 (35) 
The accrued depreciation will be 
a (1 +2) — !] 
An = 100 [oon = TL (39) 
The remaining or present value at the time when the expect- 
ancy is e years will be 
+ iy —1 
I rc 
100 100| I ha (40) 
The current annual depreciation when the expectancy is e 
years will be 
i; | (1 + gyererl ] 
dp = 1007 i TH ay (41) 
The remaining value of any perishable article depends upon 
the cost, all circumstances considered, of replacing it when it 
* For a demonstration of error in the ordinary formule see p. 456.