102 VALUATION, DEPRECIATION AND THE RATE-BASE 
useful life of which, when new, was » years, with ¢ the annual 
renewal requirement, and with g the average annual investment 
in extensions, the formulae are: 
For m less than #, 
c=" (17) 
anion 
For m greater than #, 
c= gos (-8) 
Rilia 
For very large values of m in relation to »# (n being the years 
of probable usefulness), the value of this expression approaches 
E which is the mathematical equivalent of the Straight Line 
Method. 
However desirable it might otherwise appear to introduce a 
method of computing the replacement requirement by recourse 
to amortization tables, to do this satisfactorily, in the case of a 
complex plant, is usually out of the question, when past earn- 
ings have been inadequate to accomplish the desired amortiza- 
tion. In such cases the use of some formula, as above noted, 
for estimating the probable replacement requirement is to be 
recommended and its application would be equitable from the 
standpoints of both the owner and the rate-payer. 
The Interest Bearing Annuity and the Replacement Require- 
ment. — When an annuity, bearing interest compounded annu- 
ally, is allowed to accumulate in a fund to retire invested capital, 
the demand upon the rate-payer is in annual installments. It 
would be equally proper to make the demand upon the rate- 
payer for the replacement of each individualized article, at the 
full cost of replacement, at the time when the article is discarded. 
In this case the lump sum cost of replacement, equivalent to the 
amount of the annuity, will take the place of the installments. 
If the annuity installments are forthcoming as they are due, 
then the annuity method is adequate. If the owner of the 
property does not get them, recourse should be had to the lump 
(1.