which is the formula used for computing the azimuth when the latitude of the observer, the altitude, and the declination of the observed object are known.

36. Directions.-The order of procedure for observing an altitude azimuth is as follows: Measure the altitude of the selected body and note the time shown by the chronometer at the instant of observation. At the same instant, the bearing of the object and the heading of the ship by the same compass should be noted by a second observer. Find the Greenwich date. Correct the declination of the object for the Greenwich date and find the polar distance. Reduce the observed altitude to true by applying the usual corrections. Then calculate the azimuth by the formula of the preceding article. The azimuth thus found should be reckoned from north when the latitude is south, and from south when the latitude is north, and toward east or west, according as it is A. M. or P. M. at ship.

To find the deviation, proceed similarly as in the case of amplitudes. Draw a figure and lay off the true azimuth and the compass bearing, and find the whole error. Then, using the same diagram, apply the total error to true north and lay off, respectively, the compass and the magnetic north.

EXAMPLE 1.-On December 17, 1899, A. M., while a ship was heading S W, the observed altitude of the sun's lower limb was 20° 29′ 30′′. Index error = +4'40". Height of eye = 21 feet. At the instant of measuring the altitude, the sun's bearing by the ship's compass was S by WW, and the corrected Greenwich time was December 17, 2h 3m. Latitude in = 41° 22′ N. Required, the true azimuth and the deviation, assuming the variation of locality to be 18° W.

SOLUTION.-Proceed according to directions. Thus,

In this case, since the latitude is north and the observation is made in the forenoon, the azimuth is named south and east. To find the deviation for heading, proceed as usual.

Thus,

=

EXAMPLE 2.-On December 10, 1899, at 2h 12m 36s P. M., local mean time, the observed altitude of the sun's lower limb was 34° 54′ 20′′. Index error = - 3' 44". Height of eye 28 feet. The sun's bearing by compass at the instant of observation was S 30° W. The ship was heading W S W. Latitude in = 20° 15′ N. Longitude in 161° 15′ W. Variation by chart: 20° E. Find the true azimuth and the deviation for heading.

SOLUTION.

1=

E

37. Simultaneous Observation for Azimuth and Hour Angle.-It should be observed that in the altitudeazimuth method exactly the same quantities are used as in the computation for hour angles in longitude problems; namely, latitude, altitude, and polar distance. In practice, it is therefore customary and very convenient to calculate the

azimuth at the same time as the hour angle. The compass bearing, or compass azimuth, being taken at the same time as the observation for hour angle, the same altitude may be used in determining the longitude, the true azimuth, and the deviation. Examples showing hour angle and azimuth worked out together will be given in Sumner's Method.

38. The altitude-azimuth method is applicable to any celestial body. As a rule, the object selected should be relatively low in altitude (from 20° to 40°). The compass bearing is then more readily taken and the conditions in general are more favorable for a reliable azimuth.

TIME-AZIMUTH METHOD

39. Explanation.-When the horizon is obscured so that altitudes cannot be taken, the time-azimuth method may be used for finding the true azimuth and the error of the compass. The method consists in taking the bearing of

a celestial object by compass, at the same instant noting the time by the chronometer, and from these data calculating the true azimuth by a formula derived as follows:

In the spherical triangle ZPS, Fig. 18, the colatitude ZP, the polar distance PS, and the hour angle ZPS are known,

while the altitude and the zenith distance SZ are unknown. To find the angle at Z, representing the azimuth, Napier's analogies are applied, the case being to find the angles at S and Z, two sides and the included angle being known. Thus,

Substituting in these formulas the angles Z and S, Fig. 18, for A and B, and likewise the corresponding sides and third angle for those of the figure,

The sum of these angles is the greater of the two angles Z and S, and the difference is the smaller angle. Since the greater angle is opposite the greater side, the azimuth Z is the greater angle when the polar distance is greater than the colatitude; and the smaller angle, when the polar distance is less than the colatitude. Furthermore, the angle Z is either the azimuth or the supplement of the azimuth, depending on the point of the horizon from which it is reckoned.

40. Directions. When making observation for the time-azimuth method, proceed as follows: Take several bearings in quick succession and note the chronometer (or watch) at each. Use the mean of the bearings as the compass bearing, and the mean of the times as the chronometer time. Find the Greenwich date, correct the declination for this date, and obtain the polar distance. From the Greenwich mean time, find the hour angle of the observed body by applying the longitude in time and the equation of time. If the object is the sun, its hour angle is equal to the local apparent time; if any other object, the difference between the right ascension of the observer's meridian and the right ascension of the object is the hour angle of the object.