now let 7 be the latitude of 7, or the value of the angle ARq; then 8q2Rq sin. 7 and 8 RR q2 cos.2 l. From these equations we obtain R q2 = S b2 (a2x2), and also = a2 sin.2 l و equating these values of Rq2, we obtain b4 x2 a1 cos.2 l' (a-a2x2) cos.2 l = b2 x2 sin.2 l ; a2 or, substituting a2-a2e2 for its equal b2 (by conic sections) we get after dividing by a2, Now the angle qXM (ACB) is the measure of the spherical angle APB which is the difference between the longitudes of м and q; and if this angle, in seconds, be denoted by P we shall have evidently, for the value of the arc M 9, Xq. P sin. 1", or a cos. l PROP. VII. 418. To find the ratio between the earth's axes from the length of a degree on the meridian combined with the length of a degree of longitude on any circle parallel to the equator. The radius of curvature in the direction of the meridian being represented by R, we have R 3600 sin. 1", or (art. 414.) a (1 − e2) 3600 sin. 1′′ ; or again, developing the denominator (1 - e2 sin.27) by the binomial theorem, a (1-e' + ğe2 sin. 7) 3600 sin. 1”, for the length of a degree of the meridian: and supposing one extremity (more properly the middle point) of this measured degree to have the same latitude as the parallel on which the other degree is measured, the length of the latter will (art. 417.) a cos. l 3600 sin. 1′′ or after development, be expressed by by a cos. (1 e2 sin.27) (1+e2 sin.2 7) 3600 sin. 1". Let the first of these arcs be represented by A and the second by a'; then, if the latter be subtracted from the former, we shall have ▲ — ▲' = {a — a cos. l-ae2+3ae2 sin.2 l — ae2 cos. l sin2 l} 3600 sin. 1′′; whence or A-A- -a (1-cos. 7) 3600 sin. 1′′ (a sin.2 l—a—a cos. l sin.2 7) 3600 sin. 1′′ = e2, (a sin.2 1+ a sin.2 l-a-la cos. 7 sin.27) 3600 sin. 1′′ A—A'—2 a sin.2 17 3600 sin. 1′′ 2 = e2: =e2; or again, (a sin.2 / sin.2 1—a cos.2 1) 3600 sin. 1′′ neglecting e2 in the above value of A we may, as an ap proximation sufficiently near the truth, put when we shall have A-A-2A sin.2 17 = A 3600 sin. 1" A cos. 1-A' for a; e2 = A sin.2 7 sin.2 11—a cos.2 l A sin.2 7 sin.2 11-a cos.2 l' The value of e being thus found, the ratio of the axes may be obtained as in Prop. IV. (art. 415.): and in a similar manner, the length of a degree perpendicular to the meridian being found by multiplying the value of R" (art. 414.) into 3600 sin. 1", may the ratio be obtained from the measured length of a degree of latitude combined with that of a degree on an ellipse perpendicular to the meridian. PROP. VIII. 419. To find the distance in feet, on an elliptical meridian, between the foot of a vertical arc, let fall perpendicularly on the meridian from a station near it, and the intersection of a parallel of latitude passing through the station. Also, to find the difference in feet between the lengths of the vertical arc and the corresponding portion of the parallel circle. Let P (fig. to Prop. VI.) be the pole of the world, c its centre, м the station; and let Mp, Mq be the two arcs, as in art. 410. Since normals to the elliptical meridians PA, PB, at the points м and q, may, without sensible error, be considered as meeting in one point Q on the axis PC; it is evident that the values of cos. P and of 1-cos. P, or 2 sin.2 P will be the same whether the earth be a spheroid or a sphere, and consequently that they will coincide with the values given in art. 71. Now MQ is equal to (Pl. Trigon., art. 56.), and XMQ is equal to MTB, the geographical latitude of M; therefore, putting 7 for the latitude of M, and sub stituting for MX or qx its value MX COS. XMQ we have MQ equal to a : 1-e2 sin.27) consequently, in art. 408., putting for the semidiameter MC the length of the normal MQ, there will be obtained p q (in feet) = M p2 2 a tan. 7 (1-e2 sin.2 7); or, extracting the root as far as Again, considering Mp as a circular arc of which MQ is, without sensible error, equal to the radius of curvature, it is evident that the difference between M q and мp may be obtained from its value in art. 408., on substituting for MC the above value of MQ; and it follows that on a spheroid, 1 Mq-Mp (in feet) = Mp3 tan.27 (1 - e2 sin.2 1). The value of M q being obtained for any given parallel of spheroidal latitude, it may be reduced to a corresponding arc, as M'q', having equal angular extent in longitude, by the proportion MX M'X': Mq: M'q'. PROP. IX. 420. To investigate an expression for the length of a meridional arc on the terrestrial spheroid; having, by observation, the latitudes of the extreme points, with assumed values of the equatorial radius and the excentricity of the meridian. Let and represent the observed latitudes of the two extreme stations, a the radius of the equator, and ae the excentricity; then (arts. 414, 415.) R, the radius of curvature at a point whose latitude is l, a (1-e2 + e2 sin.2 7), or putting = 3 for sin.27 its equivalent (1-cos. 27), Let m (in measures of length) express the length of a meridional arc from the equator to the point whose latitude is 7; then dm may represent an increment of that length, and if dl (in arc. rad. = 1) represent the corresponding increment of latitude, we shall have Rdl=dm; whence a (1 e 4 -e2 21) Integrating this equation between 1 and l, corresponding to m and m', (l' — 1) — şe2 (sin. 2 l' — sin. 2 1)} = m' — m ; or putting for e2 its equivalent 2 (art. 414.) and for sin. 2 l'—sin. 2 l its equivalent (Pl. Trigon., art. 41.) we have If ' be expressed in seconds, the factor and the denominator must, each, be multiplied by sin. 1". If m' m be considered as representing the length of the meridional arc, obtained in measures of length from the triangulation, this equation may be used as a test of the correctness of the assumed values of a and e. 421. In the following table are contained a few determinations of the lengths of a degree of latitude in different regions of the earth, from which the fact of a gradual but irregular increase of such lengths in proceeding from the equator towards either pole is manifest. The value of the earth's ellipticity, deduced from the measured lengths of meridional arcs, is liable to some uncertainty within, however, very narrow limits; and the ratio of the equatorial to the polar diameter is supposed to be nearly as 301 to 300. In the appendix to the Nautical Almanac for 1836 it is assumed to be as 305 to 304; whence ɛ (the compression) would be equal to 0.00247. 422. The fundamental base line is unavoidably small when compared with the distances between the stations which form the angular points of most of the primary triangles in a geodetical survey and since, when the three angles of each triangle are actually observed, the most favourable condition is that the triangles be as nearly as possible equilateral, it follows that the distances between the stations, or the sides of the triangles, should increase gradually as the stations are more remote from the base line, till those sides become of any magnitude which may be consistent with the features of the country or the power of distinguishing the objects which serve as marks. 423. The stations whose positions have been determined by the means already described, are so many fixed points from which must commence the operations for interpolating the other remarkable objects within the region or tract of country; and, for the measurement of the angles in the secondary triangles, there may be employed such an azimuth circle as has been described in art. 104., while, for triangles of the smallest class, a good theodolite of the kind employed in common surveying will suffice. A process similar to that which has been already described, may be followed in fixing the positions of the secondary stations; and where such a process is practicable, no other should be employed, the most accurate method of surveying being that which consists in observing all the angles of the triangles formed by every three objects. Should circumstances, however, prevent this method from being followed, or permit it to be only partially put in practice, the verification obtained by observing the third angle of each triangle must be omitted; and it must suffice to obtain, with the theodolite, the angles, at two stations already determined, between the line joining those stations, and others drawn from them to the station whose position it is required to find. Frequently, also, it will be convenient, in a triangle, to make use of two sides already computed, and the angle between them, obtained by observation, to compute the third side and the two angles adjacent to it. When a side of some primary triangle, or any line determined with the requisite precision, is used as the base of a secondary triangle, or as a base common to two such triangles, and the angles contained between the sides of the triangles have been observed; the rules of plane trigonometry will, in general, suffice for the computation of the lengths of the sides. The stations may also be laid down on paper, if necessary, by a graphical construction, a proper scale being chosen from which the given length of the base line is to be taken. If at any two stations already determined, there be taken, by means of a surveying compass, or the compass of the theodolite, the bearings of any object whose position is required, from the true or magnetic meridian of those stations; the intersection of lines drawn from the stations on the plan, and making, with lines representing those meridians, angles equal to the observed bearings, will give the position of the object with more or less accuracy, according to the delicacy of the needle, |