106 
been constructed, which give answer in a convenient manner to many 
questions that may be put in connection with the matter of charges. 
From Table I. we find, for instance, if we pay 40 cents for a tie, the life 
of which is six years, making the annual charges 7.88 cents, that we can 
afford to pay 15 cents per tie in addition for a process that will lengthen 
its life to nine years ; or, if we pay 25 cents for a preserving process, 
making the initial cost G5 cents, the life of the tie must be iucreased to 
at least eleven years in order that it may not increase the annual 
charge. Or if an oak tie, lasting eight years, can be bought for 50 cents, 
it would be cheaper to buy hemlock at 30 cents, though it may last but 
rive years. 
The formula used in the construction of Table I. is that for the capi- 
talization of regular annual rents, and, by transposition, assumes the 
R 1.0 » n 
form, r—-^ q p n I Pi * u wn i°h f=rent or annual charges ; recap- 
italized rent ; 2?= rate of interest ; u =the number of years for which, the 
charge is to run; and 1.0 p=^^^-. p is taken at 5 per centum, but 
as the table is intended only for purposes of comparison, the rate of in- 
terest is really irrelevant, except to allow a statement of the amount of 
saving or increased expenditure effected. 
In this table no account is taken of the reduction in annual charges 
due to the less frequent expenditure for renewal with ties of longer du- 
ration. For this a separate calculation and addition are necessary, for 
which the figures are given in Table II. 
For the expense of renewal occurring in stated but not at yearly in- 
tervals, the formula for the capitalization of intermittent rents is appli- 
cable, which by transposition appears as r= ^ ^ - — —0.0 p. For 
each cent of such expense, then, the amount which appears under the 
year at which renewal becomes necessary should be added to the an- 
nual charge found in Table I. Example : An oak tie costing 50 cents 
in the road-bed, and lasting eight years, would make an annual charge 
of 7.74 cents ; the extra expense for renewal every eight years, at 10 
cents per tie, would increase this annual charge by 0.105 x 10=1.05 cents, 
or total annual charge equal to 8.79 cents. The same tie, by creosot- 
ing, made to last sixteen years, in order to keep the annual charge the 
same, may cost 90 cents, for annual charge due to cost of renewal 
0.012 x 10=0.42 cents. Charge on cost of 90 cents, 
8.30 cents, 
8,72 cents, the total annual charge, 
