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will be the natural tangent of the angle of retreat or advance on the first curve required to make the tangent fit.

5. A still closer adjustment would be, after determining the angle approximately as above, to find the “tangents” corresponding to it for the two curves in Table XVI. Subtract the sum of these tangents from the length of the trial line, if it culs the objective curve; add the sum, if it passes outside. With the number thus found, divide the measured amount of error for the tangent of the angle of retreat or advance, as the case may be.

6. Suppose, for illustration, that a trial tangent, bearing by needle N. 54° 30' E., is run out from stake 24.80 of a 4o curve, intending to touch 6°, but is found to cut it. Suppose further that the objective 6o curve was laid down and numbered in the direction of approach towards the 4o curve; that its P. C. is stake 25.10, and the magnetic bearing of its initial tangent S. 30° 30' W. The angle, then, between the bearing of the trial tangent and that of the initial tangent of the 6o curve, is 24°, corresponding to a distance of 400 feet on the latter curve. At stake 25.10 + 4.0 29.10, therefore, a tangent to the 6° curve would be parallel to the trial tangent. Go forward on the trial tangent, accordingly, to a point opposite 29.10, and measure the distance square across to that plus on the 6° curve. Assuming the trial tangent to be 2,500 feet long, and the amount of the miss to be 87 feet, the nat. tan. of the angle of error is 0.0.348 = tan. 2o. By the method in (4), this calls for a shift of the P. T. 50 feet ahead on the 4o curve, making the new P. T. 24.S0 + 0.50 stake 25.30, and advances the P. T. of the 6o curve to stake 29.43 of that numeration.

The method in (5), applied to this case, brings the angle of error 2° 02', instead of 2o, equivalent to a deviation of 1} feet scant in half a mile from the line corrected by the method in (4), and agreeing exactly with the correction determined by the method in (2).





REVERSED CURVES. The following problems will be useful in laying off turnouts, the adjustment of tracks near stations or shops, and the like; but reversed curves should never be used on the main line between stations, where they are both objectionable and unnecessary. Ground which allows any permissible location at all will allow straight reaches of at least two hundred or three hundred feet between curves of contrary flexure; and in every case it is worth the small additional outlay to make such a location.



REVERSED CURVE HAVING EQUAL RADII. 1. The radius R, and the perpendicular distance D, between the tangents given.

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