384 NATURE OF CAPITAL AND INCOME

In order to show that this is a correct representation let us
first take the case of semi-annual income. RR" represents one
year ; Rr represents half a year. Draw the ordinate rh. From
h draw vertically upward hq to represent a half-year’s installment
 of income, that is, half of Rs. To avoid complicating the
figure, hq is omitted; were it drawn, q would lie on the line 0Q".
Then, according to our previous representation, the capitalvalue
 will follow the curve Ohg, which forms a “tooth.” The
line Og is therefore a line drawn through the top points of two
neighboring teeth. Its direction is a first approximation to

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Fic. 43.

the direction OQ of the curve for the continuous case. This
direction Oq is the diagonal of a parallelogram formed by
producing Oh to meet the ordinate from R' in H', producing
0g to Q' and completing the parallelogram.
Since H'Q' is twice as far from O as hq (ie. RR"=2 (rR)
by hypothesis), it follows, by similar triangles (i.e. Ohg and
OH'Q'), that it is also twice as long. But hg represents a
half year’s installment of income. Hence H'Q' represents two