96 VALUATION, DEPRECIATION AND THE RATE-BASE 
invested capital until the end of the probable life term. If the 
fund resulting from the accumulation of such increments be 
applied at any earlier date, a new amortization annuity, based 
on the remaining value and the remaining life, must be com- 
puted. 
Incomplete Amortization. — It will be seen that if an amor- 
tization annuity thus determined from probable life when an 
article goes into service be deducted from the investment from 
year to year, the result will be incomplete amortization. In 
the case of an article with a probable life of forty years, the 
amortization rate thus computed at 6 per cent interest would 
be 0.6462 per cent per annum. The amount of capital returned 
in forty years would be $25.85 on each $100 of capital invested 
and there would remain $74.15 still to be made good at that 
time. 
These facts make clear the point that, whenever amortiza- 
tion in lieu of replacement is accomplished by annuities bearing 
compound interest, the appraisal for rate-fixing purposes must 
be of the entire investment without deduction of accrued depre- 
ciation. 
Second Mathematical Determination of the Replacement 
Fund. — The foregoing mathematical demonstration that the 
accumulation in a replacement fund for a plant of mature age, 
when computed by the compound interest sinking fund method, 
and actually earned should amount to a considerable sum, con- 
firms a conclusion which can be reached in a more direct way. 
In the assumed case of a plant which has a life of » years, 
and of which one-nth has been constructed each year, after » 
years there will have to be replaced one-nth thereof each year. 
Because the annual investment in the installation has been 
uniform there will be, for each dollar invested per year, a total 
investment of # dollars. 
The annual replacement after # years, for each dollar annually 
invested, will be $1. If now the annuity to replace the several 
parts of the plant in # years is a dollars for each dollar of the 
annual investment, then after » years the annual amount re-