THE THEORY OF INTEREST

number for each of the n individuals, there will be in alln (m — 1) unknown /’s.

As to the x’s, there will be one for each of the m yearsfor each of the n individuals, or mn.

As to the i’s, there will be one for each year up to thelast year, m — 1. In short there will be

n (m — 1) unknown /’s,mn unknown x’s,m — 1 unknown i’s,

or 2 mn -\- m — n — 1 unknown quantities in all. Com-paring this number with the number of equations, wesee that there is one more equation than the number ofunknown quantities.

This is accounted for, as in the simplified case, by thefact that not all the equations are independent. Thismay be shown if we add together all the equations ofthe fourth set, and substitute in the numerators of thefractions thus obtained their value as obtained from thethird set, namely, zero. We shall then evidently obtainthe first equation of the third set. Consequently we mayomit any one of the equations in the last two sets. Therewill then remain just as many equations as unknownquantities, each independent (that is non-derivable fromthe rest), and our solution is determinate.

In the preceding analysis, we have throughout as-sumed a rate of interest between two points of time ayear apart. A more minute analysis would involve agreater subdivision of the income stream, and the em-ployment of a rate of interest between each two succes-sive time elements. This will evidently occasion no com-plication except to increase enormously the number ofequations and unknowns.

[298 ]