it occupies position 1, Fig. 1, with reference to the cross-hairs; the time, vertical angle, and horizontal angle are noted. Then the upper plate is loosened, the instrument turned 180° in azimuth, the telescope inverted, and the sun .sighted again, as in position 2. In position 1, the sun is moving toward both hairs; in position 2, the telescope should be set approximately as shown by the dotted circle, so that the sun will clear both hairs at the same instant. For afternoon work, the positions shown in Fig. 2 should be used. The observations are taken in pairs; if the second observation of a pair cannot be obtained promptly after the first one (owing to a passing cloud, or some other cause), the first must be ignored and considered as useless. It should be noted that the reversal of the transit between the observations eliminates the index error of the vertical circle, the error of level in the horizontal axis of the telescope, and the error of collimation of the telescope. By sighting in diagonal corners of the field of view and taking the mean of the observations, the corrections (both horizontal and vertical) due to the semi-diameter of the sun are eliminated. To simplify the notes, 180° should be added to (or subtracted from) the horizontal plate reading when the instrument is inverted. EXAMPLE. The following measurements were taken in the manner just described. The four means of the circle readings were formed in the field. The declination of the sun was -9° 30' 5", and the approximate latitude +39° 57'. Find the azimuth of the reference mark. To find the azimuth of the sun: z=71° 25' 29"; $=39° 57' 0"; 8=-9° 30' 5"; (++8)=50° 56' 12"; (+-8)=60° 26' 17". Substituting these values in the formula for the azimuth of the sun. cos 50° 56' 12" sin 60° 26' 17" sin a= sin 71° 25′ 29′′ cos 39° 57′ The two values of a are 60° 17′ 15′′ and 119° 42′ 45′′ (= 180°-60° 17′ 15′′). As the observations were made in the afternoon, the obtuse angle should be used. This gives a=2X119° 42′ 45′′=239° 25′ 30′′. The mean of the four horizontal readings is 101° 12' 8". Subtracting this from the azimuth of the sun, the azimuth of the reference mark is found to be 239° 25′ 30′′-101° 12′ 8' = 138° 13′ 22′′. RAILROAD SURVEYING DEFINITIONS OF CIRCULAR CURVES The line of a railroad consists of a series of straight lines connected by Each two adjacent lines are united by a curve having the radius curves. best adapted to the conditions of the surface. The straight lines are called tangents, because they are tangent to the curves that unite them. Railroad curves are usually circular and are divided into three general classes, namely, simple, compound, and reverse curves. A simple curve is a curve having but one radius, as the curve AB, Fig. 1, whose radius is AC. A compound curve is a continuous curve composed of two or more arcs of different radii, as the curve CDEF, Fig. 2, which is composed of the arcs CD, DE, and EF, whose respective radii are GC, HD, and KE. In the general class of compound curves may be included what are known as easement curves, transition curves, and spiral curves, now used very generally on the more important railroads. A reverse curve is a continuous curve composed of the arcs of two circles of the same or different radii, the centers of which lie on opposite sides of the curve, as in Fig. 3. The two arcs composing the curve meet at a common point or point of reversal M, at which point they are tangent to a common line perpendicular to the line joining their centers. Reverse curves are becoming less common on railroads of standard one of its extremities, is equal to one-half the central angle subtended by the chord. Thus, the angle EBC = ECB = BOC. 4. An angle not exceeding 90° having its vertex in the circumference of a circle and subtended by a chord of the circle, is equal to one-half the central angle subtended by the chord. Thus, the angle GBH, whose vertex B is in the circumference, is subtended by the chord GH and is equal to one-half the central angle GOH, subtended by the same chord GH. 5. Equal chords of a circle subtend equal angles at its center and also in its circumference, if the angles lie in corresponding segments of the circle. Thus, if BG, GH, HK, and KC are equal, BOG=GOH, GBH = HBK, etc. 6. The angle of intersection FEC of two tangents of a circle is equal to the central angle subtended by the chord joining the two points of tangency. Thus, the angle CEF = BOC. 7. A radius that bisects any chord of a circle is perpendicular to the chord. 8. A chord subtending an arc of 1o in a circle having a radius = 100 ft. is very closely equal to 1.745 ft. ELEMENTS AND METHODS OF LAYING OUT A CIRCULAR CURVE The degree of curvature of a curve is the central angle subtending a chord of 100 ft. Thus, if, in Fig. 4, the chord BG is 100 ft. long and the angle BOG is 1°, the curve is called a one-degree curve; but if, with the same length of chord, the angle BOG is 4°, the curve is called a four-degree curve. The deflection angle of a chord is the angle formed between any chord of a curve and a tangent to the curve at one extremity of the chord. It is equal to one-half the central angle subtended by the chord. The deflection angle for a chord of 100 ft. is called the regular deflection angle, and is equal to one-half the degree of curvature. The deflection angle for a subchord-that is, for a chord less than 100 ft.-is equal to one-half the degree of curvature multiplied by the length of the subchord expressed in chords of 100 ft. The length c of a subchord or of any chord is given by the formula in which c=2R sin D R=radius; D= deflection angle of that chord. Relation Between Radius and Deflection Angle. From the formula just given, с R= 2 sin D If D100 is the deflection angle for a chord of 100 ft., then For a 1° curve, D100 = 30′ and R=5,730, nearly. For curves less than 10°, 5,730 the radius may be taken as in which De is the degree of curvature. The Dc accompanying table gives the length of the radius, in feet, for degrees of curvature ranging by intervals of 5' and 10' from 0' to 20°. Tangent Distance.-The point where a curve begins is called the point of curve, and is designated by the letters P. C.; and the point where the curve terminates is called the point of tangency, and is designated by the letters P. T. The point of intersection of the tangents is called the point of intersection; it is designated by the letters P. I. The distance of the P. C. or P. T. from the P. I. is called the tangent distance, and the chord connecting the P. C. and P. T. of a curve is commonly called its long chord. This term is also applied to chords more than one station long. If I denotes the angle of intersection and R the radius of the curve, then the tangent distance T=R tan I Laying Out a Curve With a Transit.-When the angle of intersection I has been measured and the degree of curve decided upon, the radius of the curve can be taken from the Table of Radii and Deflections or it can be figured by the formula R= 5,730 The tangent distance is then computed and measured back on each tangent from the P. I., thus determining the P. C. and P. T. Subtracting the tangent distance from the station number of the P. I. will give the station number of the P. C. Ordinarily, this will not be an even or full station. The length of the curve is then computed by dividing the angle I by the degree of curve, the quotient giving the length of the curve in stations of 100 ft. and decimals thereof. After having found the length of the curve, compute the deflection angles for the chords joining the P. C. with all the station points; set the transit at the P. C.; set the vernier at 0, sight to the intersection point, and turn off 0 5 68,754.94 .145 .073 3 10 34,377.48 .291 .145 5 2,750.35 3.636 1.818 01,910.08 5.235 2.6186 50 55 0955.37 10.467 5.2349 7.846 14 0637.27 15.692 20 506.38 19.748 9.874 0 410.28 24.374 12.187 10 405.47 24.663 12.331 0359.26 27.835 13.917 0319.62 31.287 15.643 7 successively the deflection angles, at the same time measuring the chords and marking the stations. The station of the P. T. is found by adding the length of curve in chords of 100 ft. to the station of the P. C. FIG. 1 If the entire curve cannot be run from the P. C. on account of obstructions to the view, run the curve as far as the stations are visible from the P. C. and run the remainder of the curve from the last station that can be seen. Suppose that' in the 10° curve shown in Fig. 1 the station at H, 200 ft. from the P. C., which is at B, is the last point on the curve that can D be set from the P. C. A plug is driven at H and centered carefully by a tack driven at the point. The transit is now moved forward and set up at H. As the deflection angle EBH is 10° to the right, an angle of 10° is turned to the left from 0 and the vernier clamped. The in strument is then sighted to a flag at B, the lower clamp set, and by means of the lower tangent screw the cross-hairs are made to bisect the flag exactly. The vernier clamp is then loosened, the vernier set at 0, and the telescope plunged. The line of sight will then be on the tangent IP, and the deflection angles to K and C can be turned off from this tangent, and the stations at K and C located in the same manner that the stations at G and H were located from B, because the angle at IHB between the tangent IH and the chord BH is equal to the angle EBH between the tangent EB and the same chord. This method of setting the vernier for the backsight when the instrument is moved forwards to a new instrument point on the curve is sometimes called the method by zero tangent. The essential principle of the method is that the vernier always reads zero when the instrument is sighted on the tangent to the curve at the point where the instrument is set, and the deflection angles are made to read from the tangent to the curve at this point in the same manner as if this point were the P. Č. of the curve. Tangent and Chord Deflections.-Let AB, Fig. 2, be a tangent joining the curve BCEH at B. If the tangent AB is prolonged to D, the perpendicular distance DC from the tangent to the curve is called a tangent deflection. If the chord BC is prolonged to the point G, so that CG=CE, the distance GE is called a chord deflection. If the radius R of the curve and the length of the chord care known, the tangent deflection f can be determined by the formula B FIG. 2 When the two chords preceding the station considered are of unequal lengths, the chord deflection= where c1 is the length of the first chord • 2R |