208 NATURE OF CAPITAL AND INCOME [Cmar. XIII

represented by horizontal lines, and vertical lines or ordinates
 are employed — not to represent income, but rate of
income. Thus, in Figure 3, AC is the rate of income flowing
 at the instant 4, and BD is the rate at the instant B.

D

 

Fig. 3.

As a consequence, the income which flows through any
period of time, AB, is represented by the area ABDC. It
is therefore the time AB multiplied by the average rate.
In the case of a uniform flow of income, CD reduces to a
horizontal straight line. The area method will hereafter
be used wherever continuous income is in question.’

§ 4

Actual examples of true perpetual annuities cannot be
said to exist; but for practical purposes, some government
“rentes’’ are perpetual annuities, also railway leases for
999 years. While the capital value of a perpetual annuity
of $4 a year, capitalized at 4 per cent, is $100, that of an
annuity of $4 a year for 50 years is $85, for 75 years is $94,
for 100 years $98, and for 200 years $99.96. It will be seen,
therefore, that for any ordinary rate of interest, an annuity
extending a century or more is practically equal in value to
a perpetual annuity.
Among non-monetary examples may be cited as perpetual
 annuities the before-mentioned water rights in the
West. These are often sold by the miner's inch, i.e. a
theoretically perpetual flow of 1} cubic feet per minute,

1 For a fuller statement of the area method, as related to the line
method, see Appendix to Chap. XIII, § 5.