lamps, placed at the focus of a parabolic reflector, or behind a lens at the focus of parallel rays. The most perfect is the Drummond light, which may be seen at the distance of seventy miles. The best signal when the sun shines is one employed at present on the Coast Survey of the United States, called a heliotrope. It consists of a common telescope, mounted upon a three-legged stand, horizontally. It is accompanied by a man called a heliotroper, who directs the telescope to the tent in which the great theodolite is placed. Upon the eye end of the telescope is supported a small plane mirror, which has motion round both a vertical and horizontal axis, so as to be capable of being placed in a position to reflect the sun in any direction, at pleasure. The heliotroper attends and turns the mirror continually, so as to reflect the sun in a direction parallel to the axis of the telescope, which he accomplishes by causing the rays to pass through two perforated discs, supported like the mirror, on the top of the telescope tube, one being near the object end, and the other between it and the mirror.* The signal pole is supported by a wooden tripod at least its height, and the heliotrope is placed at a short distance from it, in a line with the theodolite station. All the signals visible from the station at which the great theodolite is placed, are observed every day for several weeks. The instrument is then moved to a new station, and by this means all the angles of every triangle are repeatedly observed. REDUCTION TO THE CENTRE OF THE STATION. It sometimes happens from the nature of the signal employed, that the theodolite cannot be placed at the axis of the signal called the centre of the station. The process of determining what the observed angle would have been with the instrument so placed, from observation with the instrument placed at a short measured distance from the proper point, is called reducing to the centre of the station. Let c in the diagram be the centre of the station, o the place of the instrument. From the observed angle AOB required the angle ACB. Make AOB = w, BC = 9, AC = d, ACB =x,OC=r, COB = y. Then AIB = w + IAO, and AIB = x + CBO The heliotroper is on duty till 10 A.M. and after 3 P.M., the atmosphere in the middle of the day being too unsteady near the earth's surface for good observations. If the centre of the station be inaccessible, this distance must be calculated from measurements which can be made by methods which the student will easily devise for any given case. which is the formula for the correction to be applied to the observed angle w, to obtain the required angle x. The distance r being small, d and g are computed with the angle w. VERIFICATION OF THE OBSERVED ANGLES. One third of the The excess of the sum of the angles of a spherical triangle over two right angles is technically called the spherical excess. spherical excess being subtracted from each of the angles of a spherical triangle occupying but a small portion of the surface of the sphere, a plane triangle may be formed with the resulting angles, and with rectilinear sides, equal in length to the curvilinear sides of the spherical triangle, and A, B, C being the angles, s the area of the triangle, and r the rad, of the earth (or better the rad. of curvature, see App. VI. p. 366), *This is Legendre's theorem, the demonstration of which, by Lagrange, is as follows:-The sides of a spherical triangle being a, b, and c, and the radius of the sphere r, the similar triangle on the sphere, whose radius is 1, will have for sides a b c TTT and similar expressions obtain for cos B, cos y, sin y .. (1) becomes (1) which is the formula of verification, the last term being expressed in seconds. If the sum of the three observed angles exceed 180° by observations are correct. gasin 1" the As the spherical excess is expressed in terms of the area of the triangle, that must be computed, which may be done with sufficient accuracy by considering the spherical triangle as plane, and applying to it one of the formulas of Art. 19, App. I. The angles of the plane triangle, whose sides are equal in length to those of the given spherical triangle, being found by means of the spherical excess, as above explained, and the length of one of the sides of the Transferring (1-8-y?) to the numerator, by giving it the exponent developing as far as to terms of the fourth degree inclusive, (2) becomes a1 + ß1 + y1 — 2a2 ß2 — Qa2 y2 — Qß2 y2 COS A= B2 + y2 — a2 + 24BY Replacing the values of a, ß, y, (3) may be expressed as follows: If the same lengths, a, b, c, be sides of a plane triangle, and a' be the angle opposite a, by Art. 69, COS A' = b2 + c2 — a3 Raising both members of (5) to the square, and substituting 1-sin a' for cos A' — 4 b2 c3 sin2 a′ = a1. 1 + b1 + c1 — 2a2 ba — 2a2 c2 — 2b2 c2=N Let A='+x, x being the difference between the spherical and plane angle, COS ACOs a' cos x- - sin a' sin x cos A' √1 — sin2 x sin A' sin z Putting, which is very small, for sin x, and rejecting the second power of x, COS A COS A' - sin a' Combining (6) and (7), which have the same first members (7) x= bc sin A' and since A= spherical triangle being known, that of the others may be obtained by the solution of the plane triangle. Between the latitudes 450 and 250 the spherical excess amounts to about 1" for an area of 75.5 square miles. To find the spherical excess in seconds of space therefore divide the area in square miles by 75.5. The logarithm of the mean radius of the earth in yards is 6.8427917, of r is 13.6855834. The following example shows the form* in which the above rules are applied in practice on the U. S. Coast Survey. But [Art. 19, (1) App. I.], bc, sin a' is the area of the plane triangle of which a, b, c are the sides, which area does not differ sensibly from the proposed spherical triangle. If s denote the area of either of these (8) becomes And this last is the formula of verification. The same theorem has been extended to a spheroidal triangle, the difference between the spheroidal excess and spherical excess being less than of a second in the largest triangle ever measured on the sur 20 8 face of the earth. To express in seconds, it must be divided by sin 1".t In the form given the 1st, 3d, and 4th columns explain themselves; the 2d column contains the names of the stations at the vertices of the triangle, the 7th the spherical excess 5.67, calculated by the formula on p. 323, the 5th the difference 0.15 between the excess of the sum of the observed angles over 1800, and the spherical excess, which difference ought to be zero. This error 0.15 is equally distributed between the three angles; the 6th column contains the observed angles thus corrected; the 8th column contains the angles of a plane triangle whose sides are of the same length with the spherical one, each determined by subtracting the spheri+ A still more accurate formula, deduced rigorously from the spheroid, is 1+ e2 cos 2 L in which Lmean latitude, and a = equatorial radius. This form would only become important in carrying forward azimuths in a long chain. Excess = 8 The latitude and longitude of a few of the stations of a chain of triangles, selected at different parts of the whole, being determined by astronomical observations, the latitudes and longitudes of the other stations may be found geodetically, as it is termed, by methods which we now proceed to explain. The problem to resolve is the following. Given the latitude and the longitude of the point s in the diagram, and the azimuth of the point s' upon the horizon of s*, to find the latitude and longitude of this second point, and the azimuth of s upon the horizon of s'. If L denote the given latitude, di the difference of latitude between the two stations to be applied as a correction to the known latitude, in order to find the required latitude, м the longitude, dm the difference of longitude of the two stations, z the azimuth, dz the difference of azimuth; then the following will be the FORMULAS FOR COMPUTATION OF L, M, Z OF PRIMARY TRIANGLES. For the difference of latitude - di = KB COS z + K2 c sin2 z+ (d1)2 D− h K3 E sin2 z‡ cal excess, or 1.89 from the corresponding spherical angles; this column also contains the sides of the plane triangle, the last two of which are computed from the first, and the angles by the sine proportion, p. 62, the logarithms for which purpose are contained in the last column. * Observed astronomically by methods to be described in the sequel. + The following is the demonstration of this formula:-There are known in the triangle PSS' the colatitude rs = 90°. -L of the point s, the angle PSS' the azimuth = |
