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To calculate the position of the vertex: log sin 56° 52 = 9.92293

log sin 37° 49= 9.78756 log cos 37° 49 = 9.89761

log tan 56° 52' = 10.18527 log cos pe= 9.82054

log cot (L4-L,) = 9.97283 Po=48° 35'

L-Ly=46° 47 Hence, the latitude of the vertex=48° 35' N and its longitude =46° 47' +122° 30=169° 17' W. Ans.

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.

SOLUTION.-See Fig. 16.

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Whence, A(initial course) = 89° 8'+27° 54'=117° 2=S 62° 58' E, and B(final course) =89° 8' — 27° 54'=61° 14'=S 61° 14' E To calculate the distance:

log cos 33° 56'= 9.91891 log sin 93° 59'(=cos 3° 59') = 9.99895 log cosec 117° 2'( =sec 27° 2') = 10.05025

log sin D= 9.96811

D= 68° 19
Whence, Dist. =180° — 68° 19'=111° 41'=6,701 mi.
To calculate the position of the vertex:
sin 62° 58=9.94975

sin 35° 15'= 9.76129
cos 35° 15'=9.91203

tan 62° 58'=10.29221 log cos P,=9.86178

log cot (L4-L,)=10.05350 Py=43° 20

L1-L.)=41° 29 Hence, latitude of vertex=43° 20 N. Longitude=41° 29'+75° 30 = 116° 59 w. To find the point of intersection at equator, use formula 8. Thus,

log sin 35° 15'= 9.76129

log tan 62° 58' = 10.29221 log tan (L.-L.) =10.05350

L.-L=48° 31' Whence, C=75° 30–48° 31'=26° 59' W.

Thus,

To find the course at the equator, use formula 9.

log sin 62° 58'=9.94975 log cos 35° 15'=9.91203 log cos S=9.86178

S=43° 20

D. Long. 6

D. Long. a

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Fig. 17 Hence, the course when crossing the equator should be=90° — 43° 20' =S 46° 40' E, or SEE, nearly.

To find the rhumb courses, use Mercator's sailing. Thus,

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D. Long. a=2910

D. Long. b=2729 m= M. D. Lat.=2249 m'=M. D. Lat. =2154 log 2910= 3.46389

log 2729= 3.43600 log 2249= 3.35199

log 2154= 3.33325 log tan A=10.11190

log tan B=10.10275 A=52° 18

B=51° 43 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 maximum separation in north latitude, use formulas 6 and 7. Thus, log cos 43° 20=9.86176

log sin 23° 11'=9.59514 log sin 52° 18'=9.89830

log cot 52° 18'=9.88812 log cos px=9.96346

log cot(Lr-L,)=9.70702 Px=23° 11'

Lr-Ly=63° 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 maximum separation in south latitude, use formulas 6 and 7. Thus,

log cos 43° 20=9.86176

log sin 22° 5'=9.57514 log sin 51° 43'=9.89485

log cot 51° 43'=9.89723 log cos Px=9.96691

log cot (L:-L,1)=9.67791 Px=22° 51

(Lx-Lys)=64° 32' Hence, the latitude of the point of maximum separation=22° 5' S. Since the longitude of the southern vertex (v Fig. 16) differs 180° from the northern, the longitude of maximum separation in the southern hemisphere is evidently equal to (180° – 116° 59' =) 63° 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 GREAT

CIRCLE 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

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on a flat sheet of paper X Y in such a position that one of its poles Ptouches the central part of the paper. To an observer situated at the center 0 of the globe, the latitude parallels mn and rs 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 ( upon the plane XY, 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 R=rX cot Lat., where r is a constant value

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West I East

Fig. 19 and equal to the radius of the 45th parallel. By considering r equal to unity, a still simpler formula is obtained, viz., R=cot Lat.

73. Relation Between Latitude and Radius of Parallel.—The relations existing between the latitude and radius of any parallel when projected on a chart according to

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