13. The difference of latitude and departure given to find the course and distance. EXAMPLE. From latitude 27° 15' S a ship sails to latitude 24° 39′ S, making 79.5 miles of easterly departure. Required the course steered and the distance made by the ship. SOLUTION. In this case the course and distance being sought, we use the following formulas: tan C=Dep. D. Lat.; Dist. = Dep. Xcosec C. FIG. 7 The nearest D. Lat. and Dep. in the tables are 155.9 and 79.4; the corresponding course and distance is therefore N 27° E and 175 mi., respectively. Ans. 14. The method of ascertaining a ship's position by plane sailing is sufficiently accurate in low latitudes, but only partially so in higher latitudes, unless the distance sailed is small and the course made good lies near the north and south points. EXAMPLES FOR PRACTICE 1. A ship sails from latitude 37° 30′ N to latitude 46° 8′ N, steering a course N E by N. Required the distance run made. Ans. and the departure 2. A ship from latitude 50° 13′ sails between S and E, 98 miles until her departure is 82 miles. Find the course steered and the latitude in. Ans. JCourse S 56° 47′ E. 3. A ship from St. Helena, in latitude 15° 55′ S, sails SSE E until she has made 115 miles of departure. the distance run. Find her latitude in and 4. A ship from latitude 55° 30′ N steers a course S W by S during 20 hours, and there finds by observation that her latitude in is 53° 17′ N. (a) How many knots did she make per hour and (b) what is the departure made? Ans. 5. From Cape Hatteras in latitude 35° 15′ N a N and E 226 miles, until her departure is 198 miles. and latitude in. Ans. (a) 8 knots per hr. (b) Dep. =88.9 mi. ship sails between Find her course f Course = N 61° 11' E. Lat. in=37° 4′ N. 6. A ship runs S E by E from latitude 1° 45′ N, and then by observation is in latitude 31' S. Required her distance and departure. Ans. [Dist. =244.8 mi. Dep.=203.5 mi. TRAVERSE SAILING 15. In the examples of sailings hitherto introduced, the student will notice that only a single course has been considered. But since a vessel may change her course several times during a voyage, or in the interval of a few hours, it becomes necessary to have a method, or process, by which the several courses and distances run for a certain interval of time may be reduced to a single course and distance, which the vessel would have made had she gone directly from the place left to the place arrived at. This method or process is called working a traverse, and problems relating to this method are usually treated under the head of Traverse Sailing. 16. Explanation and Principles.—In order to elucidate the principles of traverse sailing, the student is referred to Fig. 8. Suppose a ship to have sailed from 4 to B, from B to C, from C to D, and from D to E; NS and WE representing, respectively, the directions of the meridians and parallels. Then, by laying out the difference of latitude and departure for each course, a number of small right-angled triangles are formed, and by dropping a perpendicular from A to the parallel KE, a larger right-angled triangle A KE is formed, in which the side A Krepresents the total difference B of latitude and KE the total departure corresponding to the total distance A E, and the course KAE made good by the ship. Now, by examining the figure it will be seen that the total difference of latitude A K, and the total departure KE 274 561 161 i.......i........ FIG. 8 are, respectively, the algebraic sum of the several differences of latitude and departures made on each of the courses sailed by the ship. To prove this, let easterly departures and northerly differences of latitude be denoted by (+), and westerly departures and southerly differences of latitude be denoted by (-). We then have, and, -Aa+C'C-nD+Dm=KE, +a B-BC'+Cn+mE=AK. Now, by substituting the numerical values of the different departures and differences of latitudes, according to the scale attached to Fig. 8, and taking their algebraic sum, we get the whole departure KE=-16+41-171+25=321; and the whole difference of latitude AK=+27-15+241 +20=56 miles. The accuracy of this may be verified by actual measurements on the figure. 17. Constructing a Traverse. From the foregoing it is evident that in order to find the total departure and difference of latitude made good by a ship that has sailed on several courses, it is necessary to find the departure and difference of latitude corresponding to each separate course and distance. This is found either by calculation or by the traverse tables, and the several departures and differences of latitudes thus found are then entered into a table constructed of four columns headed N, S, E, W, respectively, similar to the one shown below. The figures in each column are then added together, when the difference of the sums of the N and S columns will be the total difference of latitude made good and is given the same name as the greater sum; likewise, will the difference of the E and W column be the total departure made good to be named the same as the greater sum. From this total departure and difference of latitude is then calculated the course and distance made good according to formulas of Plane Sailing. Problems in traverse sailing may be solved either by the traverse tables or by construction, which will be shown in the following illustration. In practice, however, the method by traverse tables is the one more commonly used. 18. Illustration.-The following courses and distances are sailed: S W by S, 24 miles; N N W, 57 miles; S E by EE, 84 miles; and south, 35 miles. Find the course and distance made good. Proceed as follows: Construct a traverse table consisting of six columns; the first column to contain the courses; the second, the distances; the third and the fourth, the D. Lat. made good = 41.9'S 39.0' E = Dep. made good. differences of latitude; the fifth and the sixth, the departures. The table having been constructed, enter the traverse tables and take out the difference of latitude and departure for each course and distance run, and insert them in their proper columns. Thus, when the course is southerly, the difference of latitude must be entered in the column headed S, when northerly in column headed N; the departure when westerly must be entered in the column headed W; and when easterly, in column headed E. For the beginner, a good idea is to make a horizontal dash in all places where no figures are to be entered, as shown in the table. The columns properly filled up, add the differences of latitude in each column and |