minutes, if any, in I to decimals of a degree; then divide by the degree of curvature.

Example 1.

I - 20° 40′, D = 3°. 20° 40′ is equivalent to 20.67 degrees. Dividing by 3, we have 6.89 chord lengths for the length of the curve. If the chords, as is usual, are each 100 feet long, the length of the curve in this case will be 689 feet. If the chord lengths were 50 feet each, the length of the curve would be half this number of feet.

14. If the degree of curvature is fractional, the more convenient method of effecting the division is, first, to reduce both I and D to minutes; then divide the former by the latter.

I = 30° 22′, D=2° 45'. to 1,822 and 165 minutes.

Example 2.

These are equivalent, respectively,
Dividing the former by the latter,

we have 1,104 feet for the length of the curve.

15. The ingenious assistant who will attentively consider the preceding figures cannot fail to detect other obvious analogies which it has not been thought necessary to include in this compendium.

16. In railroad field practice it is usually sufficient to determine angles to the nearest minute, and distances to the nearest foot. The nicety of seconds and tenths appears generally to be quite superfluous; the time consumed on them were better employed in pushing ahead.

If the chords are 100 feet long, as is usual in railroad prac

5,730, the radius of a 1° curve, by the deflection angle, or degree of curvature. Thus, in the foregoing example, 5,730 ÷ 4 = 1,432.5.

=

18. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE DEFLECTION ANGLE D.

From the preceding equation and example:

C÷R=50 ÷ 1,432.7 = .0349 = sin 2o = D

19. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE DEFLECTION DISTANCE d.

First find the deflection angle by above method (18). Then, angle HAB in the figure being made equal to D, and II A = BA, BH will be the deflection distance. Draw A K to the middle point of HB.

=

Then d =HB2 KB=2AB X sin KAB = 2 C X sin D.

Example.

Let R= 1,146 feet, C = 100 feet.

By (18) D will be found = 5o.

Then d = 2 C × sin D = 200 × .0436: = 8.72 feet.

20. If the chords are 100 feet long, as is usual in field measurement, divide the constant number 10,000 by the radius in feet: the quotient will be the deflection distance. The deflection distance with radius of 10,000 feet and chord of 100 feet is one foot: this rule is based upon the principle that deflection distances, the chord length being fixed, will vary inversely as the radii.

Thus, in the foregoing example, 10,000 1,146 = 8.72.

21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TANGENTIAL ANGLE T.

The angle T is equal to D by construction; for mode of

22. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TANGENTIAL DISTANCE t.

Then,

=

First find the tangential angle, as above directed. angle BAE in the figure being made equal to T, and AE AB, BE will be the tangential distance. Draw AN to the middle point of BE.

Then t 2 CX sin T = 200 × .0218=4.36 feet.

23. In ordinary railroad practice the tangential distance may be considered equal to half the deflection distance.

XIX.

ORDINATES.

1. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE MID

In the annexed figure, HN = M, H G = R, AB = C.

NG AGAN2R2-C2; HN-HG-NG,

Example.

R=819, C=100; to find the middle ordinate, M.

AGB.

HN =

AN X tan. HAN; i.e., M = being the central angle subtended by the chord.

CX tan. D; D

3. GIVEN THE RADIUS R, CHORD C, AND MIDDLE ORDINATE
M, TO FIND ANY OTHER ORDINATE E K = M', DISTANT ₫
FROM N, THE MIDDLE POINT OF THE CHORD.
KL=NG; NK = = GL; EKEL-NG.

ELGEN KR-d; NG (1)=√R2 — C2. Then E K=M'R' -d-/R2 — C2.

4. It is a property of the parabola, that ordinates vary as the products of their abscissas. This property may be assigned to the circle in cases where the arc encloses a small angle. Applying it here we have

HN EK: ANX NB: AK X KB.

Call any segments A K, K B, of the chord, a and b.

Then MM: : 4 C2 : ab, . '. M'

== MX 4 ab÷C2.

Example.

M = 1.528, C = 100, a = 60, b = 40; to find M'.

M' 1.528 X 9600 10000 =
=

1.528 X 0.96 = 1.467.

5. Multiply the corresponding ordinate of a 1° curve from the annexed table by the degree of curvature: the product will

ORDINATES OF A 1° CURVE, CHORD 100 FEET.

DISTANCES OF THE ORDINATES FROM THE END OF THE 100-FEET CHord.

Middle

Feet. Feet. Feet. Feet. Feet. Feet. Feet.

Feet. Feet. Feet.

What is the ordinate of a 6° curve, 30 feet from the end of the 100-feet chord?

The corresponding tabular ordinate of a 1° curve is .183; which, multiplied by 6, gives 1.098, the required ordinate.

6. A quick way of laying off ordinates on the ground, and one sufficiently exact for the field, is, after fixing the point II by means of the middle ordinate HN, to stretch a line from H to A, and make the middle ordinate FOHN; then from F to A and F to H, making the middle ordinates FO; and

so on.

7. A good track-layer will seldom require points at shorter intervals than 25 feet.