124 
(4). Annual amount to be allowed (./.• cents) for cost of changing tie 
after n years is : 
0.05 
l.U5-i-1.05 X " CeUtS - 
Taking the value of old wooden tie at 10 per cent, of first cost, that of 
old iron sleeper at 40 per cent., it would be found that a wooden tie of 
61 first cost and 12 years' life costs 10.2 cents per annum exclusive of 
maintenance, while an iron tie of $2 first cost, lasting 30 years, costs 
9.8 cents per annum.* 
Another writer inquires how much more a metal tie will cost than a 
wooden one, if the former would last double the time of the latter and 
its value as old material be considered 40 per cent, of its first cost 
(O /= =0.4 N,), while that of the old wooden tie is taken as 10 per cent, of its 
first cost (O /y =0.1 S /f ). The cost of renewal (B=]S — O) is, in the case 
of the metal tie, R 7 =0.6 N /? and for the wooden tie E 7/ =0.9 N„. 
The annual charge for renewal (r), if life of ties is set at n, and n N 
vears. and the rate of interest at p per cent., mav be expressed: 
(— ! 5S l — ) 
g t 
r.=0.6 X, for metal 
e n > — 1 
^=0.9^ ^ — for wood, 
and these amounts capitalized (that is to say, the capital E, the interest 
on which at p per cent., will furnish the required amounts of yearly 
charge for reserve fund), 
E = would make 
e—1 
B -_M_N\ for metal. 
' e—i 
^ 0.9 „ „ 
B.,=— =N.. for wood. 
" e—1 
Leaving out all other advantages of iron ties, that is to say. consider- 
ing only the longer life as the criterion, X' may be taken so high that 
the first cost of the metal tie. together with the capital necessary to re- 
new it indefinitely, equals the sum of the same items for wooden ties, 
namely : 
N / +B / =K // +B /i or N.^+B^-B^ 
and by substitution 
- '-(e»,_0.4) (e"«- 1) _ /; 
If, theu, n,=2 n „i that is to say. if the metal tie last double the time of 
he wooden tie, the cost of the first may be 
_e 2w »+0.9e B «-0.1 y 
"See table on page llr : ai»> ln>no n of page 122. 
