I

an estimate of the error for places on the same side of the equator may be formed by the help of a few cases. Suppose the course 4 points or 45°, and the difference of latitude 10° or 600'; then if this difference of latitude is made good in any latitude below 30°, the error of the difference of longitude will not exceed 2'; if made good between the parallels of 40° and 50°, the error will be about 3'; and between 60° and 70° about 19', or of a degree. For smaller distances the errors will be much less, and for greater distances much greater, as they vary in much more rapid proportion than the distances. It has been observed before that when the course is large, the difference of longitude should be found by middle latitude in preference to Mercator's sailing; because, although the latter is mathematically correct in principle, yet a small error in the course may, when the course is large, produce a considerable error in the difference of longitude. The reason of this is easily shown. In middle latitude sailing we convert the departure into difference of longitude. The process increases the departure in a proportion which is less than 2 to 1 in all latitudes below 60°; and exceeds 3 to 1 in all latitudes beyond 70°. The error of the departure, increased in the same proportion, becomes thus the error of difference of longitude. Now when the course is nearly E. or W., the departure is nearly the same as the distance, and an error of some degrees in the course does not affect the departure sensibly; hence in this case the error of the difference of longitude depends on that of the distance alone. But in Mercator's sailing, on the other hand, we convert the meridional difference of latitude into difference of longitude, and the process, when the course is large, converts a given meridional difference of latitude into a difference of longitude much greater than itself; and thus increases the error of the meridional difference of latitude in the same proportion. Thus, for example, at the course 80°, the difference of longitude exceeds the meridional difference of latitude in the proportion of 6 to 1; at the course 85° this proportion is 11 to 1. Now, when the course is large, a slight change in it sensibly affects the difference of latitude, and also the meridional difference of latitude, which is deduced directly from it. In high latitudes the meridional parts vary rapidly, and the error of the difference of longitude is increased accordingly; hence the precept more especially demands attention in high latitudes."-Raper's Practice of Navigation, pp. 103, 104.

THE DAY'S WORK.

284. This is the process of finding the ship's place at noon-that is, its latitude and longitude, having given the latitude and longitude at noon preceding, or a departure taken since, the compass courses and distances run in the interval, the leeway (if any), variation and deviation (if any), direction and rate of current (if any), &c., &c.

RULE LXXIII.*

1°. Correct each course for leeway, variation, and deviation (see Rules XLVIII to LV, pages 120 to 160, which arrange in the tabular form as in the example following. Add together the hourly distances sailed on each course, and insert the same in the Table, opposite the true course.

NOTE.-Allow the leeway in points before expressing the course in degrees.

Departure Course.-When a departure has been taken, consider the opposite to the bearing as a course, which correct for variation, and the deviation due to the direction of the ship's head when the bearing was taken, and insert in the Table as an actual course, with the distance of the object as a distance. The departure course is generally put down in the Tables as the first course. See No.

251, page 157.

As the ship leaves the land, the bearing (by compass) of some prominent object or known headland is taken, and its distance is generally estimated by the eye; this process is called "taking a departure." The latitude and longitude of the landmark are known; and thence, by supposing the ship to have sailed on a course the opposite to the bearing of the object, through the distance that object is off, we thus obtain, on commencing a voyage, a determinate starting point, from whence to reckon the subsequent courses and distances. Thus, supposing, for example, a ship leaving the Tyne observes Tynemouth Light dipping, and setting it, finds its bearing to be W. by N., distant (by estimation) 20 miles. Now in sailing from Tynemouth Light to the present position of the ship, she would have to sail in the opposite direction to the bearing of the light, viz., E. by S., 20 miles. At the end of the day, the Day's Work gives us a change of the ship's place as referred to the landmark, and not the supposed position. For methods of determining the distance, see Raper's Practice of Navigation, on Taking Departures, ch., IV, pp. 114-122.

Current Course.-The set of a current is to be corrected for variation only (being correct magnetic), and inserted in the Table as a course; the drift being taken as a distance. The current course is generally inserted in the Table as the last course.

2°. Take out of the Traverse Tables (Table I or II, RAPER or NORIE) the difference of latitude and departure to each course and distance, (see Rule LXIII, page 184), and proceed to find the difference of latitude and departure made good as directed in Rule LXV, page 189, Traverse Sailing.

3°. Find the course and distance made good (see Rule LXIV, page 186.) 4. Find the latitude in by applying the difference of latitude to the latitude from (see Rule XLIV, page 108).

If a departure has been taken, the difference of latitude is to be applied to latitude of the point of land; if otherwise, to yesterday's latitude.

Nearly the entire process of computing the Day's Work has already been given, and if the learner has thoroughly mastered the rules laid down in the preceding pages, he will find no difficulty in working the Day's Work without reference to them.

NOTE.-When the course is less than 5 points or 56°, the difference of longitude may be found by either or both Middle Latitude or Mercator's method, but if the course exceeds 5 points the method of Middle Latitude should be used in preference to Mercator's (see Remarks in pages 204-205).

5°. To find the difference of longitude.-By Middle Latitude Sailing. (a) Find the middle latitude as directed, Rule XLV, page 108.

(b) Next at the page of Traverse Table on which the degrees (at top or bottom) correspond to middle latitude, find the departure in a difference of latitude column, then the corresponding distance is the difference of longitude of the same name as the departure (see Rule LXX, 4°, and note, page 197.)

(c) Or thus, by calculation:-To log. sec. of middle latitude add log. of dep., the sum (rejecting 10 from the index) is the log. of diff. of long.

When the latitude left and latitude in are of contrary names, that is, in low latitudes, no sensible error can arise from taking the departure itself as the difference of longitude.

6°. If the ship has made a due E. or due W. course good, the difference of longitude is found by Parallel Sailing, thus:

With the latitude as a course and the departure in a difference of latitude column, then the corresponding distance is the difference of longitude (see Rule LXVII, page 194).

7°. To find the difference of longitude.—By Mercator's Sailing.

(a)

Find meridional difference of latitude (see Rule XLIII, page 107). (b) Then with course and meridional difference of latitude (in a latitude column), find the corresponding departure, which is the difference of longitude (see Rule LXXII, page 203).

(c) To find the long. in.—With the longitude left and difference of longitude find the longitude in (see Rule XLVII, page 110).

When a departure has been taken the longitude left is that of the point of land; otherwise that of yesterday.

EXAMPLE I.

The corrected courses are written down to the nearest degree, and the work will stand as follows:

Previous to opening the Traverse Table to take out the difference of latitude and departure to each course and distance in the above table, fill up the columns not wanted: thus, in the first course, S. 60° W., the S. and W. will be wanted, and the N. and E. will not be wanted; fill up these last two columns by drawing a dash under N. and E. Proceed in the same manner with the other courses.

2. To find the difference of latitude and departure to each course and distance by the Traverse Table.

Enter Traverse Table, and take out the difference of latitude and departure corresponding to 60° and distance 21'. Insert them in the columns S. and W.

The second course is S. 19° E., and the distance 22'7. Then, 19 degrees and distance 227 (omitting the decimal point) give difference of latitude 214'6, departure 73'9; now dropping the tenths in each, namely, the 6 and the 9, and shifting the decimal point one place to the left, we have difference of latitude 215, departure 7.4, which insert in columns S. and E., the course being marked S. and E.

The third course is N. 14° W., and distance 146. Look for 14 degrees and distance 14'6, which give difference of latitude 141'7, departure 35'3; now dropping the tenths, the 7 and the 3, and increasing the preceding figure by 1, in the first case, as the tenths exceed 5, we have, by removing the decimal point one figure to the left, the difference of latitude 142, and departure 3.5.

Proceed in this way with the remaining courses.

EE