APPENDIX TO CHAPTER XII 363 $100 will to-day buy the right to $4 a year forever, and a year hence this right to $4 annually may be sold on a 3%, basis. It will therefore fetch $133.33. The investor will consequently receive in all, at the end of this first year, a total of $4 4 $1331, or $1371. Hence the rate of interest in the sense of a premium will be for that year 374%. In succeeding years the rate of interest, considered either as a premium or a price, will evidently be 3%. § 5 (ro Cum. XII, § 6) Mathematical Relations between the Rates of Interest as a Premium and as a Price In general, if we let ¢, represent the rate of interest in the premium sense for this year, i, for next year, 4; for the third year, and so on, whereas jy, j, and js etc., represent the rates of interest in the price sense for the same successive years, the following relations between the ¢’s and j’s may be proved: — 1 1 1 : re fe roy sf eee + oo 84 ADE ==1+ A+) A+’ 1 1 4 + +. sodint Fr 3 =<+ E : : . 144 A+a)A+ih) A+a)A +a) +h) 1 1 1 Tt dL 144; A+4)* A+4)° 1 1 1 : ~t 2 = + g = ——-... ad inf. 144 (A+i)A+id) A+i)A +i) +15) 1 1 1 — tt + 144s (+44) Q+0)° : 1 1 : - - de Jind int. +a Armd +n A+ud+od+i) ete. These equations may be said to determine jj, as a peculiar sort of mean of the magnitudes 4, ta #, - - - ad inf., and ja 28 a similar mean of 4, fy 4 - - - ad inf., ete. Their proof, which is simple, is left to those readers who are interested in mathematics. ai The preceding equations express the values of the j’s in ad inf. = . ad inf. =