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ExAMPLE 3.-Calculate the great-circle track between Cape Hatteras, latitude 35° 15' N and longitude 75°30' W, and Cape of Good Hope, latitude 53° 56' S and longitude 18°29' E.
Hence, the rhumb course from A to C- S 52° 18' E, and from B to C-N 51° 43' W.
To calculate the point of marimum separation in north latitude, use formulas 6 and 7. Thus,
Hence, the latitude of the point of maximum separation in the northern hemisphere=23° 11' N, and its longitude=116° 59'-63°= 53° 59' W.
To calculate the point of marimum separation in south latitude, use formulas 6 and 7. Thus,
Hence, the latitude of the point of maximum separation=22° 5' S. Since the longitude of the southern vertex (z' Fig. 16) differs 180° from the northern, the longitude of maximum separation in the southern hemisphere is evidently equal to (180°-116° 59' =) 6.3° 1' E-64° 32' = 1° 31' W. Ans.
NotE.—In this case the advantages gained by following the greatcircle track would be inconsiderable, since the distance on the rhumb line is but 44 miles longer than great-circle distance, and the difference between the rhumb course and the initial and final course is only 1 point. By following the great-circle track between San Francisco and Tokio, Japan, in example 2, not less than 271 miles is saved. The rhumb course between these places is S 88.4° W, hence the difference between it and the initial and final course is more than 3 points.
GRAPHICAL METHODS RELATING TO Gre ATCIRCLE SAILING 71. Great-Circle Charts.-Charts constructed on the gnomonic projection are called great-circle charts, and on such charts the great-circle track between any two places is represented as a straight line connecting them.
72. The principles of the gnomonic projection are illustrated in Fig. 18. A globe representing the earth is placed
on a flat sheet of paper X Y in such a position that one of its poles P' touches the central part of the paper. To an observer situated at the center O of the globe, the latitude parallels m n and r s will now appear on the paper as concentric circles described around the center P', and the meridians will appear as straight lines radiating from the same center. Supposing the meridians on the globe to be 10° apart, and to be projected from the center O upon the plane X Y, the result will, upon the removal of the globe, appear as in Fig. 19, which now represents a chart on the gnomonic projection. - The difference of longitude between any two places, for instance that between A and B, is now measured by the angle at the pole P’ subtended by the meridians passing through the two places, and similarly the difference of longitude between any two places is measured by the angle at the pole. The radius R for each latitude parallel is obtained from the formula A' = r) cot Lat., where r is a constant value