370 NATURE OF CAPITAL AND INCOME 
of the capitalized annuity if the annuity is payable annually, 
semi-annually, or quarterly. If the annuity be $4 a year, the 
teeth drop $4 if payable annually, $2, if semi-annually, and $1, 
if quarterly. If the frequency of the installments of income be 
2 
Annual Payments 
  
2 2 2 2 
2 3 4 
Semi-annual Payments 
LALA ad 
1 2 3 
Quarterly Payments 
Fic. 34. 
indefinitely increased, we reach the limiting case of a continuous 
income, when the teeth disappear entirely and the value of the 
annuity remains at a constant level. The value of the annuity 
of $4 in all these cases, just after any installment of in- 
come is received, will be $100, if the rate of interest 
is 49, provided the rate is respectively reckoned annually,” 
semi-annually, and quarterly in the various cases. In the 
case of continuous income, the value of the annuity of $4 
will always be $100 if the rate of interest be 49, “reckoned 
continuously.” The same remarks apply, of course, to an i 
annuity of any number of dollars. Its value after each install- 
ment is equal to the annual income divided by the rate of in- 
terest, or the annual income multiplied by the purchase period. 
In order to obtain the formula for the value of a perpetuity 
payable semi-annually, quarterly, or continuously, in terms of i, 
the rate of interest reckoned annually, we need only to transform 
> 2 . by means of the equations in the previous chapter. Thus