364 NATURE OF CAPITAL AND INCOME terms of #s. The following equations give the ¥’s in terms of the y's: — h=n+ Lids, Jz i= Jat Jad ds, Js iy=js +L , J4 ete. Thus if, as in our example, j, =.04 and j,=.03, . 04 —.03 ry ih =.04 + — =.37%. The proof of these formule is also left to the mathematical reader. He will observe that the two sets of equations may be proved independently or either set may be proved and the other set derived from it. To show that either set may be derived from the other, it will be found useful to substitute for the left-hand members of the first set their simpler values as derived by algebra. These are, ho ete. An easy proof of J1 J2 J3 this is found by actually dividing 1 by 7, ete. Fron the formule it is clear thatif 4, = i, = iy =, ete., then jh =js=Js=, etc., and that then all the ©s = the j’s. The converse is also evident. § 6 (ro Cu. XII § 7) Mathematical Relations between the Rates of Interest and Discount Let V', due one year hence, be the equivalent of V available in the present. Then the rates of interest and discount are expressed respectively by the formula: — ! 1+i=353 V 1-d=7 Whence, by multiplying the two equations together we derive (1414) (1—d)=1, which reduces to d=1i—id. That is, the number representing the rate of discount equals the number representing the equivalent rate of interest less a