IN TERMS OF FORMULAS

of those sets. Thus, if we add together all the equationsof the fourth set, we get the first equation of the thirdset. ( namely, x-f + x 2 + x 3 ' — 0). The addition gives

(x/ + x/ + x 3 ') + x ” + x f + x *- = 0.

In this equation we may substitute zero for the nu-merator of the fraction (as is evident by consulting thesecond equation of the third set). Making this substitu-tion, the above equation becomes

Xi + x/ + x 3 ' = 0,

which was to have been proved. Since we have here de-rived one of the five equations of the last two sets fromthe other four, the equations are not all independent.Any one of these five may be omitted as it could be ob-tained from the others. We have left then only ten equa-tions. Since no one of these ten equations can be derivedfrom the other nine, the ten are independent and arejust sufficient to determine the ten unknown quantities,namely, the /’s, a^’s, x"’s and i.

§7. Case of m Years and n Individuals

We may now proceed to the case in which more thanthree individuals (let us say n individuals) and morethan two years (let us say m years) are involved. Weshall assume, as before, that the x’s, representing loansor borrowings, are to be considered of positive value whenthey represent additions to income, and of negative valuewhen they represent deductions.

§8. Impatience Principle A (n(m — 1) Equations )

The equations expressing Impatience Principle A willnow be in several groups, of which the first is:

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