from C to A. From the meaning of longitude we have GNA=3° 4′, GNC = 63° 35′, whence α= ANC = 60° 31′. Also, by the meaning of latitude, AN=90°— 53° 24′ = 36° 36'; CN=90° 44° 40′ = 45° 20'. We therefore have, in the spherical triangle CNA, CN= a = 45° 20', AN= c = 36° 36′ and the included angle α = 60° 31', which is case 3 in the solution of spherical triangles. The data: = 1⁄2 † Therefore the bearings of C from A are N. 77° 26' W.; of If only the distance sailed is required it is simpler to use the law of cosines. Thus, cos x = cos c cos a + sin c sin a and the distance = 39.4° × 69.1 = 2723 miles. 87. The course of the ship at C would be N. 54° 54' E. To show how the ship's course changes as it sails along CA let us find the course as the ship crosses the meridian 38° W. at the point B, Fig. 35. In the triangle NCB we have b= CN 45° 20', C= NCB = 54° 54', N: CNB = 63° 35' — 38° = 25° 35'; that is, two angles and the included side. 1⁄2 (C+N) = 40° 14.5', (CN) = 14° 39.5', b = 22° 40'. 1-44 log sin (CN) = 9.4033 log tan b = 9.6208 (C+ N) = 0.1897 (c — n) = 9.2138 1 (c + n) = 27° 53.5' 1⁄2 (c — n) = 9° 17.5'. Whence c = BN = 37° 11', n = CB = 18° 36'. The latitude of B is 90° – BN = 52° 49′ N., and the distance sailed is CB = 18.6° × 69.1 miles = 1285 miles. To find the angle B = CBN we have log sin (c + n) = 9.6700 log tan 1⁄2 (C — N) = 9.4176 log csc 1 (c — n) = 0.7919 log cot B 9.8795 } B=52° 51', B = 105° 42', and NBA = 180° – B = 74° 18′. Therefore the ship's course at B (the bearing of A from 88. The Area of a Spherical Triangle may be found as follows: The theorem has been proved that the area of a spherical triangle is equal to its spherical excess (the excess of the sum of its angles over two right angles) times the area of the tri-rectangular triangle; it being understood that the right angle is the unit of angles. Thus, using A to represent the area of a triangle whose angles (in degrees) are A, B, and C; and noting that the tri-rectangular triangle is one eighth of the surface of the sphere; we have Example. Given A= 105°, B = 80°, C = 95°, and taking 3960 miles, the radius of the earth, EXAMPLES In the following problems assume that one can travel directly along the arc of a great circle between the points named. 1. A ship sails from Baltimore to Boston. Find the course of the ship as she leaves Baltimore, her course as she crosses the meridian of New York, and the entire distance she sails. What are the bearings of Baltimore from Boston, and of Boston from Baltimore ? 2. Find the course at Liverpool, the course at 55° W., and the total distance sailed by a ship going from Liverpool to New York. What are the bearings of these cities from each other? 3. A ship sails from Baltimore to Rio de Janeiro. She sails first to a point off Pernambuco in latitude 8° S., longitude 34° W., and from there to Rio. How far does she sail, and what is her course off Pernambuco ? NOTE. In the Southern Hemisphere latitudes are taken as algebraically negative. Use the north-polar distances of places as sides in solving triangles. 4. In problem 3 what course will the ship be sailing after she has gone 1000 miles? What will be her latitude and longitude at that point? 5. How far is the Washington Observatory from the Greenwich Observatory? What are the bearings of the two observatories from each other? 6. A ship sails from Boston on a course East 12° South. At what distance would she be sailing due East? What are her latitude and longitude at that instant? 7. A ship sails northwest from San Francisco. What would be the highest latitude she would reach? What would be the ship's longitude at that instant ? 8. Find the number of square miles in the triangle whose vertices are Baltimore, Boston, and Chicago. 9. A ship sails from Honolulu to San Francisco. Find the entire distance sailed, and the course of the ship when she has gone halfway. 10. An aëroplane sails from Washington to Chicago along a great circle arc one mile above the surface of the Earth. In what time is the flight made at a rate of 75 miles per hour ? 11. Find the number of square miles in the triangle whose vertices are Baltimore, New York, and Chicago. 12. Find the face and edge angles of a regular triangular pyramid. 13. What is the latitude of three points on the Earth equally distant from each other and from the North pole? 14. Each face of a triangular pyramid is a triangle whose sides are 3, 4, and 5 respectively. Find the face and edge angles of the pyramid. |