To find the difference of longitude. Middle lat. track; sum of the lats. of the extremities of the (225) and, by (224), Diff, long. = departure cos, mid. lat. dep. X sec.mid. lat. (226) or, by substituting (212), Diff. long = dist. X sin, course X sec. mid. lat. (227) “ This method of calculating the difference of longitude may be rendered perfectly accurate by applying to the middle latitude a correction,” which is given in the Navigator, and the method of computing which will be explained in the succeeding chapter. [B. p. 76.] 33. By combining the triangle (fig. 15) of Plane sailing with that (fig. 18) of Parallel sailing a triangle (fig. 19) is obtained, by which all the cases of Middle Latitude sailing may be solved. 34. Problem. To find the distance and bearing of two places from each other, when their latitudes and longitudes are known. [B. p. 68.] Solution. From (fig. 19) we have Departure = diff. long. X cos. mid. lat. (228) Cases of middle latitude sailing. 35. Problem. To find the course, distance, and difference of longitude, when both latitudes and the departure are given. [B. p. 70.] Solution. The difference of longitude is found by (226), the course by (229), and the distance by (230). 36. Problem. To find the departure, distance, and difference of longitude, when both latitudes and the course are given. [B. p. 72.] Solution. The departure is found by the formula departure = diff. lat. X tang. course ; (231) the distance by (230); and the difference of longitude may be found by (226), or by substituting (231) in (226) diff. long. = diff. lat. X tang. course X sec. mid. lat. (232) 37. Problem. To find the course, departure, and difference of longitude, when both latitudes and the distance are given. [B. p. 73.] Solution. The course is found by the formula diff. lat. COS, course = ; dist. (233) the departure by departure = dist. X sin, course; and the difference of longitude by (226). (234) 38. Problem. To find the difference of latitude, dis 140331 Examples. tance, and difference of longitude, when one latitude, course, and departure are given. (B. p. 74.] Solution. The difference of latitude is found by the formula (235) diff. lat. = dep. X cotan, course ; the distance by the formula dep. X cosec, course; and the difference of longitude by (226). dist. = (236) 39. Problem. To find the course, difference of latitude, and difference of longitude, when one latitude, the distance, and departure are given. [B. p. 75.] and the difference of longitude by (226). 40. Examples. 1. A ship sailed from Halifax (Nova Scotia) a distance of 2509 miles, upon a course S. 79° 34' E. ; find the place at which it arrived. Solution. By $ 32, diff. lat. =454' = 7° 34' S. 2.65740 given lat. =44° 36' N. mid. lat.=40° 49' required lat. = 37° 2' N. cor. 7 cor. mid. lat. =40° 56' sec. 10.12178 = diff. long. 3.51404 given long. = 63° 28' W. required long. 9° 2' W. Ans. The place arrived at is one mile south of Cape St. Vincent in Portugal. 2. Find the bearing and distance of Canton from Washington. Solution. By $ 34, lat. of Washington=38°53' N. long.= 77° 3' W. lat, of Canton =23° 7' N. long.=113°14' E. diff. lat.=946'=15°46', sum of longs.=190°17' 31' cor. = COS. cor, mid.lat, 31° 31' diff. long.=10183 diff. lat. 946' 9.93069 ar. CO. 2.97589 bearing =S. 83°47' W. tang. 10.96268 sec. 10.96524 3.94113 dist. =8732 miles Examples. 3. A ship sails from New York a distance of 675.4 miles, upon a course S. E. 1 S.; find the place at which it arrives. Ans. Three miles to the west of Georgetown in Bermuda. 4. Find the bearing and distance of Portland (Maine) from New Orleans. Ans. The bearing N. 49° 24' E. The distance = 1257 miles. 5. A ship from the Cape of Good Hope sails northwesterly until its latitude is 22° 3' S., and its departure 3110 miles ; find its course, distance sailed, longitude, and its distance from Cape St. Thomas (Brazil). Ans. Course = N. 76° 38' W. = 3197 miles. Longitude = 40° 36' W. 6. A ship sails from Boston upon a course E. by N. until it arrives in latitude 45° 21' N.; find the distance, its longitude, and its distance and bearing from Liverpool. Ans. Distance sailed = 922 miles. Longitude 50° 10' W. Distance from Liverpool = 1905 miles. Bearing from Liverpool S. 75° 22' W. 7. A ship sails southwesterly from Gibraltar a distance of 1500 miles, when it is in latitude 14° 44' N.; find its course and longitude, and distance from Cape Verde. |