358 NATURE OF CAPITAL AND INCOME out for $1, which he may evidently do if the rate of interest remains unchanged. With his total receipts, 1 + he buys a new annuity of the same type. This will evidently yield him WU (1 3 ¢' per annum, payable in semi-annual installments of (+3) half that amount, or i At the end of another six months, then, he receives this last-named sum, and, Selling his newly bought annuity for its original value of 1 + 7 he has in ovidy ) hand a total sum of 1 4% LE Of this sum he reini, a (1 T BD vests $1 and retains as income the remainder, or 2 he 3 This sum may evidently be obtained year after year simply by repeating the above process. It constitutes a perpetual annuity, payable annually, and its value simplified from the $12 above formula is evidently '+ Since this is the annual income, payable annually, which $1 of capital will buy, it is by definition, the magnitude we called ¢; pr) that is, T= +3. We may in = manner proceed to quarterly payments, in which case we shall find, by analogous reasoning, denoting by i" the rate of interest per annum payable quarterly, that yy Be gm ge lr TT § 2 (ro Cu. XII, § 4) Mathematical Relations between Rates reckoned Annually, Semi-annually, etc., when Rates are conceived as * Premiums.” Diagrammatic Representation. Economic Interpretation of e. In general let ¢' be the rate of interest per annum reckoned semi-annually. Then the “amount” of $1 in six months is