116 VALUATION, DEPRECIATION AND THE RATE-BASE
similar to that which has above been cited as more probable
than failure always at the end of the probable life term), the
fact remains that in no particular case, no matter how large a
plant may be, will there be absolute conformity with any as-
sumed law. Consequently figures determined on the basis of
any reasonable hypothesis of failures can be used to prepare
smoothed-out curves and from such curves, tables for general
use can be prepared. It is enough to know for the present
that ordinarily it may be assumed that the replacement require-
ment of a large number of articles with a probable life of »
years should increase progressively year by year to about one
ath of the cost of effecting complete replacement of all articles
and that this rate of one nth would be reached at about the
nth year.
Table 7 has been prepared to show for numerous articles, all
of which when new have a probable life of ro years, the re-
placement requirements on the assumption that failures actu-
ally occur according to the hypothesis already explained, and
to show their relation to the remaining value, the current amor-
tization, and the required annual earnings estimated by the
Straight Line and by the Equal Annual Payment methods of
procedure, also the remaining investment, the replacement allow-
ance, and the required earnings if estimated by the Unlimited
Life Method. These methods of procedure, when rates for public
utilities are to be fixed, are explained in Chapter IX.
Expectancy of an Equivalent Single Article.— When numerous
articles are under consideration, it may be desirable to know
the expectancy of an equivalent single article, representing the
aggregate of all the separate articles, which may be used to
simplify the estimates of accrued and current depreciation. If
the Straight Line Method of procedure is adopted and if nu-
merous articles of the same kind and same individual cost are
involved, and if it be assumed that actual life in each case will
agree with the estimated probable life, then the expectancy of
the equivalent single article will be the average of the remaining
ages of the individual articles. But under all other methods of
