. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . 3. Hence it appears that the sum of the errors of the obser-vations is .23—(—!) = , which the observer must 352 SPHERICAL TRIGONOMETRY. add to the three observed angles, in such proportions as hisjudgment may direct. One way is to increase each of theobserved angles by one-third of , and take the anglesthus corrected for the true angles. 229. Reduction of an Angle to the Horizon. — Given theangles of elevation or depression of two objects, which are at asmall angula
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . 3. Hence it appears that the sum of the errors of the obser-vations is .23—(—!) = , which the observer must 352 SPHERICAL TRIGONOMETRY. add to the three observed angles, in such proportions as hisjudgment may direct. One way is to increase each of theobserved angles by one-third of , and take the anglesthus corrected for the true angles. 229. Reduction of an Angle to the Horizon. — Given theangles of elevation or depression of two objects, which are at asmall angular distance from the horizon, and the angle whichthe objects subtend, to find the horizontal angle between them. Let a, b be the two objects, the angular distance betweenwhich is measured by an observer at0; let OZ be the direction at rightangles to the observers horizon. De-scribe a sphere round 0 as a centre,and let vertical planes through Oa,Ob, meet the horizon at OA, OB, re-spectively ; then the horizontal angleAOB, or AB, is required. Let ab = 6, AB = 0 + x, Aa = h,B6 = k. Then in the triangle aZb we have. cos AB = cos aZb ?. or cos (6 + x) = cos ab — cos aZ cos bZsin aZ sin bZ cos 0 — sin h sin k cos h cos k This gives the exact value of AB; by approximation weobtain, where x is essentially small, n * cos 0 — lik cos 6 — x sin 6 = — • l-i(/*2 + &2) .-. x sin 0 = hk — % (h2 + k2) cos 6, nearly. x = 2 hk - (h2 + k2) f cos21 - sin2 ^\ 2 sin 6l[(h + k)2 tan £ 6 - (h - k)2 cot £ 0]. SMALL VARIATIONS IN PARTS OF TRIANGLES. 353 EXAMPLES. 1. Prove that the angles subtended by the sides of aspherical triangle at the pole of its circumcircle are respec-tively double the corresponding angles of its chordal tri-angle. 2. If Al3 Bb Ci; A2, B2? C2; A3, B3, C8; be the angles ofthe chordal triangles of the colunars, prove that cosA1 = cos JasinS, cosB1 = sinJ6sin(S —C), cosC^sin^ csin(S — B), cosA2 = sin Jasin(S —C), cosB2 = cosJ6sinS, cosC2 = sin Jcsin(S —A
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Keywords: ., bookcentury1900, bookdecade1900, booksubjecttrigono, bookyear1902
