HIGHWAY BONDS. 103 
in the first; and in the third, by adding those in the second to those 
in the fourth as a check. 
Generalization of the annuity loan.—The preceding discussion 
can most easily be generalized by considering the loan of aq dollars 
where both principal and interest at effective rate 1 per annum are 
discharged by equal annual installments of 1 at the end of each year 
for n years. The initial principal is a,; the interest, 7a,,=1—v"; the 
annual payment, 1, of which t— (1—v") =v" is applied to repayment 
% Reid 
of principal. But @i—v"=a>q; hence the outstanding principal 
at the beginning of the second year is a=, as might have been 
predicted in advance. A repetition of this process leads to the fol- 
lowing schedule: 
ScHEDULE II.—Showing repayment of principal and interest at effective rater per annum 
on a loan of az by equal annual payments of 1 at the end of each year for n years. 
Principal Teas Der ee 
Y outstanding Interest due at Annual Dey, Pr incipal 
ear. Scone Saal Ghee ment at end repaid at 
ning of year | aka of year. end of year. 
g of year. 
1 an| l—vn if un 
2 1S l—7y"1 i Oars 
n 1] 
3 a3] 1—y" i aya 
iE Qn [—yn-kt 1 yn-kt1 
n az] 1l—v 1 v 
Totals (n—az)/2 N— Oxy | n ; az 
Since this is a schedule for a loan of az, if each item in it, apart 
from those in the column headed “year,” is divided by a” and 
multiplied by Z, there results the corresponding schedule for a loan 
of Z dollars. 
For example, the items on a loan of Z dollars for the kth year 
would be 
rae On (1 —e"s**)) /az, [an Le"—* lan. _(@8) 
There are some curious properties revealed by the above schedule, 
among which the following may be pointed out. The principal 
_ repayments on an annuity loan increase in geometrical progression, 
the factor bemg 1+2. The sum of these repayments is aq; the sum 
of the annual payments is n; the total interest is n—ay; and the 
check on the first and second columns shows that 
Denia Gra lease sus eke) - + Gn) = — Ay. 
It is apparent that most of the items in the schedule can be filled 
in directly from the a” and v" tables. Having thus filled in each 
