APPENDIX TO CHAPTER XII 367

V
But 7=1-4
therefore 1—d= (1 = 7)
2
da' 2
and a=a'—(%) .

from which we see that the discount rate reckoned annually
is less than its equivalent reckoned semi-annually. Similar
reasoning applies to quarterly and other rates.

§ 8 (ro Cm. XII, § 8)

Dimensions of the Rates of Interest, Discount, and Capitalization
The rate of interest in the price sense is of the form Vv

where v represents the part of the perpetual income stream
which flows in the time ¢, and V represents its capital value.
Since, in this fraction, » and V are of the same dimension,
both being measured in dollars, or else in bushels, or in some
other units of same denomination, the dimensionality of the

fraction > reduces to ] or t~%* In like manner, the ratio

of capitalization, being the reciprocal of this fraction, or r

has the dimensionality ¢.
These results are recognized in common usage, inasmuch as
the rate of interest is so much per annum, and the ratio of
capitalization is so many years’ purchase.
The same dimensionality is obtainable from the rate of
interest considered as a premium. The rate of interest as a

premium is given by the equation - = (141), where V' and

V are two exchangeable sums separated by the interval ¢,
usually a fraction of a year, for which the rate of interest is

to be reckoned. Thus, if the reckoning is quarterly, ¢ is 1.
!

: : Y.
Since V' and V are of the same dimension, 7 is a pure number.

e dimensionality of this ‘price of
dimensionality of the price of one
in the Appendix to Chap. L

* It is interesting to observe that th
capital» is entirely different from the
kind of goods in terms of another, as shown