SURVEYING

observed angles, and must be distributed among those angles so as to render the sum precisely 180° + E before the sides are computed. It is therefore nece ary to determine E. Let S denote the area of the triangle in square feet, r the number of feet in the radius of the earth, and!r = 3-14159; then E is given in seconds Ijy this formula (see Spherical Exce ), ^ S X 648000" E= ^72 (G48000 being the number of seconds in 180°). In order, therefore, to compute E, we must previously know the values of S and r. Now with respect to S, it is to be observed that in every case which can arise in jiractice the area of the triangle must be a very small quantity in comparison of?-2, so that in order to find E it is not nece ary to compute S with great precision. A sufficiently near value will be obtained by calculating one of the unknown sides as if the triangle were a plane one, and computing the area from the formula S = i o 6 sin. C. With respect to r, which is here taken to represent the radius of curvature of the surface of the triangle in question, it is to be remarked that by reason of the ellipticity of the earth, the radius of curvature of any arc on the earth's surface varies not only with the latitude of the place of observation, but also with the direction of the arc in respect of the meridian. For the present purpose it would be sufficiently accurate to a ume r as the radius of the meridian; but as the radius of the perpendicular and oblique arcs is required in other parts of the computation, we shall here state the formula from which they are computed. Let R be the radius of curvature of the meridian at latitude /, R' the radius of the circle pependicular to the meridian, and r the radius of a great circle making an angle = 0 with the meridian; also let p denote half the polar axis of the earth = 20,852.394 feet (see Degree), and e the ellipticity, or the difference between the equatorial and polar axes divided by the polar axis (= 1 -i- 301-026 = -(03322); then R = p (1 — e-|-3esin.2/) R'=:p (1 -(-e+ esin.2/) r = R(l + 3^^^sin.2 0) Since the inclination S is different for each of the sides of the triangle, a mean value of r may be found by making 0 = 45°, in which case sin.2fl = A; and as the curvature varies very little through a considerable extent of country, the same value of r may be used for all the triangles within a zone of two or three degrees of latitude. Suppose, then, the value to be computed for the mean latitude of a chain of triangles, the formula for the spherical ex.„ ^ ^ ah sin. C x 648000 ce will be E = —; and on a ummg the constant 648000 -v- 2 t r^ = tn, the formula for computation will be log. E = log. a + log. h + log. sin. C + log. m. It is proper to remark that in general E is a very small quantity. When the sides of the triangles are about 20 or 30 miles, it will seldom exceed 4 or 5 seconds of a degree; but in some of the great triangles connecting Ireland with the west coast of Scotland its value was found to exceed 30 seconds. Having computed the spherical exce E, make Page 1212