observed at different times. Let zx zy be their zenith distances. Then in the figure we know by observation z x and zy, and from the Nautical Almanac we can find the polar distances PX and Py; also by means of the elapsed time as measured by a watch, or from the right ascension of the bodies, or from both, we can compute the polar angle Xpy; the colatitude Pz may then be computed in the following manner by the application of the common rules of spherical trigonometry.

1. In triangle Pyx are given two sides PX, Py and the included angle X Py to find xy, which call arc 1.

2. In triangle PXy are given three sides PX, PY

1, to find angle Pxy, which call arc 2.

and arc

3. In triangle zxy are given three sides zx, zy and arc 1, to find angle z xy, which call arc 3.

4. Arc 2

arc 3 = angle P X Z = arc 4. But if the arc xy drawn through x and y pass when produced between P and

W

N

P

IQ

y

X

E

z the pole and the zenith, then it is evident by the annexed figure that the arc 2+ arc 3

PXZ or arc 4. If the arc xy produced pass near z, the bodies x and y in such a position should not be observed.

Lastly. In triangle PXZ are given the two sides PX and z x and arc 4 (namely, the in

cluded angle Pxz), to find Pz the colatitude, and thence the latitude.

Correction for run.

If the ship have moved in the interval between the observations, the second altitude will in general differ from what it would have been if both observations had been taken at the same place. On this account it is usual to apply to the first altitude a correction so as to reduce it to

what it would have been if taken at the place of the second observation; this quantity is called "the correction for run of the ship," and may be calculated as follows.

When a ship describes an arc on the surface of the sea, the zenith describes a similar arc in the celestial concave :

Z

Z1

D

D

Z

let, therefore, z be the zenith of the ship at the first observation, z' its zenith at the second observation ; then arc z z' measures the distance run in the interval. Let s be the place of the heavenly body at the first observation: with centre s at distance s z', describe an arc cutting sz, fig. 1, or s z produced in D, fig. 2; then the triangle z z'D being small, may be considered as a right-angled plane triangle, and z D. is the correction to be applied to z s in order to get z's the distance of s from the zenith at the second observation.

Fig. 1.

Now ZD

Fig. 2.

Z Z' cos. Z' Z D

= distance run x cos. angle between the direc

tion of the ship's run and the bearing of the sun at the first observation.

This correction ZD may be readily found by means of the traverse table, for since (Astronomical Problems, p. 122),

Diff. lat. dist. cos. course; if therefore in triangle ZZ'D the angle z z'd be considered as the course, and z z' the distance, the correction z D for run will correspond in the table to the difference of latitude.

The angle z'z D is the difference between the course of the ship in the interval and the true bearing of the body, when the run of the ship has been towards the place of the body, as in fig. 1; and what this angle wants of 180° or 16 points when the direction of the ship's run has been from the place of the body, as in fig. 2. In the former case it is manifest that the correction ZD for run must be

added to the first observed altitude, and in the second subtracted, in order to get the altitude of the body, the same as it would have been if it had been also observed at the place of the ship at the second observation.

Rule XXXV. (for run).

1. Enter the traverse table with the distance run as a distance, and the angle (supposed less than 8 points) between the true bearing of the heavenly body at the first observation and course of the ship, as a course, and take out the corresponding diff. lat., which add to the first true altitude (the tenths in the diff. lat. being turned into seconds, by multiplying them by 60); the result will be the altitude corrected for run.

2. But if the above angle be greater than 8 points, subtract the same from 16 points, and look out the remainder as a course, and subtract the diff. lat. corresponding thereto from the first true altitude; the result will be the altitude corrected for run.

EXAMPLES.

1. The course of the ship was N.W. W. 10 miles, and bearing of the sun E. by S., required the correction for the first altitude for run.

The angle between N.W. W. and E. by S. is 13 points, snbtracting 13 from 16 points: enter traverse table with the remainder, namely, 2 as a course and 10 miles as a distance: the corresponding diff. lat. is 8'8 = 8' 48" to be subtracted from the true altitude.

2. The course of the ship was E.N.E. 25 miles, and bearing of the sun E. by S., required the correction of the first altitude for run.

The angle between E.N.E. and E. by S. is 3 points; entering traverse table with 3 points as a course, and 25 miles as a distance, the corresponding diff. lat. 28'8 = 20′ 48′′ to be added to the true altitude.

=

(141.) The true course of the ship was S.W.W. 15 miles, and the true bearing of the sun S. by E. E., required the correction of the first altitude for run. Ans.,+5′ 42′′. (142.) The true course of the ship was W. N. 19 miles, and the true bearing of the sun was S. by E. E., required the correction for run. Ans., 7' 18".

Rules for finding the latitude by double altitude.

Rule XXXVI.

First. When the object observed is the sun.

1. From the estimated mean time at the ship at each observation, and the longitude, get two Greenwich dates.

2. By means of the Nautical Almanac find the declination for each Greenwich date. Take out also from the Almanac the sun's semidiameter.

3. Find the polar distance at each observation by subtracting the declination from 90°, if the estimated latitude and declination are of the same name; or by adding 90° to the declination, if the estimated latitude and declination are of different names.

4. Correct the two observed altitudes for index correction, dip, semidiameter, and correction in altitude.

5. Correct also the first altitude observed for the run of the ship (p. 143).

6. Subtract the true altitudes thus obtained from 90° and thus get the zenith distances.

7. Find the polar angle or elapsed time between the observations, by subtracting the time shown by chronometer at the first observation from the time shown by chronometer (increased if necessary by 12 hours) at second observation.

NOTE.-When great accuracy is required, this elapsed time should be corrected for rate of chronometer, and also for the change in the equation of time in the interval; but these corrections are seldom made.

8. To find arc 1 (using Inman's Tables). Add together

log. sin. polar distance at greater bearing, log. sin. polar distance at lesser bearing, and log. haversine of polar angle; reject 10 in the index and look out the result as a log. haversine; the arc corresponding thereto is arc 1 nearly.

9. To find arc 2. Under arc 1 put polar distance at greater bearing, and take the difference, under which put polar distance at lesser bearing; take the sum and difference of the two last quantities. Add together the log. cosecants of the two first arcs put down, and halves of the log. haversines of the two last arcs put down; the sum, rejecting 10 in index, is the log. haversine of arc 2, which take from the Tables.

10. To find arc 3. Under arc 1 put zenith distance at greater bearing, and take the difference, under which put zenith distance at lesser bearing: take the sum and difference of the last two quantities.

Add together the log. cosecants of the two first arcs put down, and halve the log. haversines of the two last arcs put down; the sum, rejecting 10 in index, is the log. haversine of arc 3, which take from the Tables.

11. To find arc 4. The difference between arc 2 and arc 3 is arc 4.

NOTE. When the arc joining the places of the sun at the two observations passes, when produced, between the zenith and pole (which the observer may easily discover at the time the observation is taken), then the sum of arcs 2 and 3 is arc 4.

12. To find arc 5. Add together log. sin. polar dist. at greater bearing, log. sin. zenith distance at greater bearing and log. haversine of arc 4, the sum, rejecting 10 in the index, is log. haversine of arc, which take from the Tables, and call arc 5.

Take the difference between the polar distances at the greater bearing, and the zenith distance at greater bearing. Add together versine of arc 5 and versine of the difference just found; the sum is the versine of the colatitude,

H