A treatise on the nature and properties of algebraic equations; with an appendix on the solution of equations by means of symmetrical functions

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1832 ... for it m x 0,000004848. The same remark applies also to last Art. (6) Equation (1) gives the correct angle between the chords; but it is not sufficiently easy of application, where many triangles are to be solved by this method. 'LEGENDRES THEOREM. 41. If from each of the angles of a small spherical triangle one-third of the spherical excess be subtracted, the triangle may be solved as a plane one, without sensible error. Let a, b, c be the sides of the spherical triangle to rad r; C one of the angles; and x the quantity to be subtracted from it, in order that it may be considered as a plane triangle. Then, c a b cos---cos-cos-r, r r r cos C= _. a. h 2 sin-sin-r r _ 2rl 24r2"' V 2r2 24r2 "A 2rl 24r "') abr w V x r2 6r2 /V 6r2 ) (by substituting for the sines and cosines their expansions, and neglecting higher powers of-&c. than the squares) a2 + b2-c2---,(2atb2 + 2a2c1 + 2b2c2-a-b-c) GENERAL ROY'S THEOREM FOR THE SPHERICAL EXCESS. 42. To find the logarithm of the spherical excess, estimated in seconds, subtract 9,3267737 from the logarithm of the area of the triangle, (taken as a plane one) in feet. For, if jc=area of triangle; r=Earth,s radius in feet, A + B + C-180" =18 60 x (Art. 36) A, B, and C being expressed in seconds, and r=mD (Art. 6. (a) Plane Trig)=--D 7T.'. excess in seconds 60 x 60 x iT 1 x. Now Tt=3, 14159, and the mean measured length (D) of a degree on the Earth's surface, in feet, = 6 x (60859, l). And by making these substitutions, reducing, and taking the logarithms, we shall have, log excess = log x--9,3267737. CHAP. IV. ON GEODETIC MEASUREMENTS. SECTION 2. 43. Geodetic measurements are employed, first, to map an extensive country; secondly, to ascertain the figure and dimensions of the Earth. The first operation...