A very large proportion of the items of expense in a train-mile are absolutely unaffected by curvature. It will therefore simplify matters somewhat if we at once throw out all the unaffected items. Of the items of maintenance of way and structures, all but Items 2 to 6 may be thrown out. Item 9 would be affected in case a bridge or a trestle occurred on the curve considered, but since the very large majority of bridges and trestles are purposely made straight, and since only a very small proportion of the total length of the curves of a road will be found on its bridges and trestles, the effect of this exceedingly small percentage on the cost of this small item would be so very minute that it may be utterly neglected.

## 165. Item 3. Renewals Of Ties

Curvature affects ties by increasing the rail-cutting and by requiring more frequent respiking, which spike-kills the ties even before they have decayed. Wellington estimates that a tie which will last nine years on a tangent will last but six years on a 10° curve. He adds 50% for tie renewals. He considers the decrease in tie life to be proportional to the degree of curve. This statement is again a verification of the general statement in § 163, that the extra cost per foot of the sharper curvature just balances the extra length of the easier curvature.

## 166. Item 4. Renewals Of Rails

It has already been demonstrated in Chapter IX (Track Economics), §§ 102 and 103, that the rate of rail wear on curves seems to bear some relation to the stage of that wear in the life-history of the rail, and also that the rail wear is nearly, if not quite, proportional to the degree of the curve. Since the fundamental feature of the method of obtaining the effect of curvature on the operating expenses is to assume that the extra expense varies as the number of degrees of central angle, we may here assume that the law is also applicable to the wear of rails. In § 103 it was computed that the excess rail wear on a 10° curve would be 226% of the rail wear on a tangent. Wellington assumed that the extra rail wear on a continuous 11° 20% curve would be 300%, which would be the equivalent of 268% extra rail wear on a 10° curve. Although the value determined above is somewhat less than Wellington's value, it is based on what are perhaps the most complete and reliable series of tests ever made on such a subject, and we will therefore assume the value 226%.

## 167. Item 6. Repairs Of Roadway

A very large proportion of the sub-items are absolutely unaffected. The care of embankments and sloped, the ditching, weeding, etc., are evidently unaffected. The track-labor on rails and ties and the work of surfacing will evidently be somewhat increased, and yet it is very seldom that the length of a track section would be decreased simply on account of excessive curvature throughout that section. We are here trying to estimate how much this item, which consists largely of track-labor, will be affected by 528° of central angle per mile. In the previous chapter an approximate estimate was made that the average curvature per mile of road for the whole United States is about 35°. 528° of curvature in a mile probably does not frequently occur. It would mean the equivalent of nearly 1 1/2 complete circles, and yet it is probably a generous estimate to say that the track-labor and other expenses belonging to this item would not be increased more than 25% for such an amount of curvature. Items 2 and 5 are also allowed 25%.

## 168. Effect Of Curvature On Maintenance Of Equipment

All items except the repairs, renewals and depreciation of steam locomotives, passenger-, freight- and work-cars, and shop machinery and tools, will be considered as unaffected. As before, electric equipment is ignored.

## 169. Items 25-27. Repairs And Renewals Of Locomotives

Curvature affects locomotive repairs by increasing very largely the wear on tires and wheels, and also the wear and strain due to the additional power required. Since the resistance due to curvature is very small compared with that due to even a moderate grade, this last cause may be neglected altogether. Referring to Table XXIII (§ 139), we find an estimate that 19% of the cost of engine repairs is assigned to curvature and grades combined. Of this amount two-thirds, or, say, 13%, should be assigned to curvature alone. On the basis that the average curvature of the roads of the country is about 35° per mile, which is about one-fifteenth of the 528° of curvature per mile which we are considering, then, if 35° is responsible for 13% increase in engine repairs, 528° would be responsible for 196%. It must be admitted that the above computation is grossly approximate, and that it contains the unwarrantable assumption that the extra cost of engine repairs which is due to curvature will be strictly in proportion to the degrees of curve. Although it is probably not true that 528° of curvature would increase the cost of engine repairs by 15 times the extra cost of 35° of curvature, yet it is probably true that for ordinary variations from that average of 35° per mile the increased cost of engine repairs will be approximately as the number of degrees of curve, and therefore our final value is not necessarily far out of the way. If 35° is responsible for an increase of 13%, 1° would be responsible for about .37 of 1%. In allowing an increase of 196% for 528° we are also allowing .37 of 1% per degree of central angle.

## 170. Items 31-36, 43-45

By a similar course of reasoning to that above given, the estimates for Items 34-36 and 43-45 will be made 100%, while that for Items 31-33 will be made only 50%, because such a large proportion of the expenses of Items 31-33 are due to painting and maintaining upholstery, which have no relation to variations in alinement.

## 171. Item 46

The repairs and renewals of shop machinery and tools will not be increased more than 50% per mile for the additional repairs required on the above equipment.