APPENDIX TO CHAPTER XIII 393

any particular number of individual rates, as dé, 4, i, ete.,
used for discounting individual items of income, a, a, a,
would evidently be,

a Ay a
en - eS er: ale a 00
wtapraipto- 



a, Ag ag
TES Oars
Various applications of such formula might be made, though
they have little practical utility. Thus, the value of a terminable
 annuity of a given term is found, as before, by taking the
difference in value between a perpetnal annuity beginning today
 and a perpetual annuity deferred to the end of the given
term. The value of the perpetual annuity beginning to-day
would be found, as has just been shown, by dividing the annual
income a by j;, the average rate of interest of the individual
rates from the present into the indefinite future. The value
of the deferred annuity taken at the end of the term would be,

 

in like manner, * where j, is the average of the individual
t
rates from that point indefinitely forward. The present value

of this deferred annuity would be found by discounting the
latter value for the term of the annuity according to the rate
h,. where jy, is the average of the individual rates of interest
iy, Tg Tg - - . 7, for the term of the annuity.
The value of a bond would be found in a similar manner.
We have just shown how to find the present value of the
“interest” of the bond, and the present value of the “prineci-7


” i idently ————.
pal,” P, due at maturity, would be evidently Tt

§ 16 (ro Cm. XIII, § 11)
Representation of Capital and Income by Polar Coordinates

The mathematical reader may be interested in an alternative
method of representing income and capital, by which polar
coordinates are employed instead of rectangular. Let the
radius vector in Figure 49 represent the parent capital. The time
required for a complete revolution of the radius vector may be