386 NATURE OF CAPITAL AND INCOME

the composition of forces or motions is very evident. Availing
ourselves of this analogy, we may say that in the case of discontinuous
 income, the point O traces the capital curve (backward)
 by obeying alternately two tendencies, — one, to follow
the discount curve at times when no income occurs, and the
other, to rise vertically whenever income occurs. In the case
of continuous income these motions occur simultaneously
instead of alternately, and the resultant is a smooth curve
instead of a series of teeth.
The same principles apply when part of the income curve
is below the horizontal axis, representing negative income. If

 

 

Fic. 45.

at the beginning the prospective cost just counterbalances the
prospective income, the capital-value will at that instant be
zero, and will from that point rise and fall again to zero at the
end, as indicated in Figure 44.
The formula for capital-value in the case of a continuous
income stream will be,—
da
=Ja+wy
in which da may be said to represent the infinitesimal income
which flows in the infinitesimal increment of time df. In other
words, da represents an infinitesimal element of the area ABC
(Fig. 45), of which element the base or breadth represents the infinitesimal
 time dt. For the purpose of integration we may substitute
 for da the expression f(f)dt, where f (f) represents CB, or
the ordinate of the income stream taken as a function of time 2.
If the special form of the function f (¢) is known, it is evidently
possible to integrate the expression and obtain its value for any
given limits,