IN TERMS OF FORMULAS

§9. Reconciling the Numbers oj Equations andUnknowns

The reconciliation of these two discordant results iseffected by two considerations. One reduces the numberof equations. Just as under the first approximation, wehave one less independent equation in the two sets ex-pressing the Market Principles than the apparent num-ber, thus making the final net number of equations

3 mn + m — n — 1.

The other consideration is quite different. It subtractsfrom the number of unknowns. This can be done be-cause each r, of which there are n(m — 1), is a derivativefrom the y’s. By definition r is the excess above unity ofthe ratio between a small increment in the y of next yearto the corresponding decrement in the y of this year.The same applies to any pair of successive years. Thisderivative is, more explicitly expressed, a differentialquotient. 3

The reader not familiar with the notation of the differ-ential calculus will get a clearer picture of the inherentderivability of the r’s from the y’s by recurring to thegeometric method in Chapter XI. There y' and y" areshown as the coordinates (“latitude” and “longitude”)of the Opportunity line, while r is shown as the tangentialslope of that line. It is evident that, given the Oppor-tunity line, its tangential slope at any point is derivedfrom it. It is not a new variable but is included in thevariation of y' and y" as the position on the curvechanges.

If, now, we subtract n{m — 1), the number of the r’s,

’See Appendix to this chapter (Chapter XIII), §1.

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