APPENDIX TO CHAPTER XIII 371

the value of an annuity of a dollars per annum payable semiannually
 is : which, by substituting for i its value as derived

from the relation between i and #, viz. 1+4=(1 +3 , or

i! = 2(V1 + i — 1) becomes g

2(vV1+4i-1)
Sree cates
4(vV1+i—-1)
annuity, since 1+4i=e¢? or 8=log,(1+7), becomes 2 GID as
In every case the value just before an installment is found by
adding that installment to the results just derived; and the
value at intervening points by applying the discount curve,
i.e. dividing the impending value (just before the next in- ii
stallment) by (1 + i), where ¢ is the time between the present "i
and the time of the next installment. lee

§ 5 (ro Cu. XIII, § 3)

Diagrams for Discontinuous and Continuous Income

Similarly, the

quarterly annuity becomes , and the continuous

 

If the income installments recur annually, and are $4 each,
these installments are represented in the line method by a, a, a,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a a a a
b lb b b b b b
|
c C c | Cc Cc Cc Cc C Cc C Cc C
Fic. 35.
i in Fig. 85. If they occur semi-annually, in installments of $2
each, they are represented by b, b, b. If they occur quarterly, in