164   NAVIGATIONAL COMPASSES

ured on the celestial meridian passing through the given body. If the polar distance and the declination are measured from the same pole, the polar distance equals 90° - declination; if they are measured from opposite poles, the polar distance equals 90° + declination.

The hour angle of the sun relative to some chosen place on the earth is called the local apparent time at the given place. Apparent time is expressed either in hours or in degrees. Thus, we may speak of the sun being a certain number of hours, minutes and seconds east (or west) of Greenwich. One hour equals 15 degrees of are. As the apparent motion of the sun relative to the earth increases and decreases in the course of a year, it is convenient to think of a fictitious " mean sun " that is assumed to have a uniform motion relative to the earth. The Civil Day begins at the instant of transit of the mean sun across the lower branch of the meridian of the observer, that is, at midnight. Clocks indicate civil time, that is, the hour angle of the mean sun, measured from the lower branch of the meridian of some selected place on the earth - Greenwich, for example. At any instant, the difference between the apparent and the mean time, that is, the difference between the hour angles of the apparent and the mean sun, is the equation of time. Equations of time are tabulated in the Nautical Almanac for every even hour of Greenwich civil time throughout the year. Knowing his longitude from Greenwich, an observer with a clock keeping Greenwich civil time can find his local civil time. Then, from the equation of time given in the Nautical Almanac, he can find his local apparent time.

Whenever the sun is visible, the compass bearing can be observed. If we know the three sides of the spherical triangle XYZ, we can compute the azimuth or true bearing 0 of the sun, measured from the north, by means of one of the standard equations of spherical trigonometry,

21 B =sin'   z)sin2(x+y+z)

cos 2    2      (114)
sin x sin y

where x, y, and z represent the sides of the triangle opposite the corners X, Y, and Z, respectively. The difference between the computed azimuth and the observed compass bearing of the sun is the angle between the north-south lines of the compass card and the geographical meridian where the compass is situated.

THE VARIOUS TYPES   165

From Fig. 127 we find the following values for the quantities in this equation:

sin '2[x + y - z] = sin 2 [(90° - latitude) + (90° - altitude) -

polar distance]

= sin [90° - 2(lat. + alt. + p.d.)]

= cos 2 (lat. + alt. + p.d.)

sin 2[x + y + z] = sin [90° - 2 (lat. + alt. + p.d.) + p.d.]

= cos (lat. + alt. 2    + p.d. _ p'd )

1 =   1   __ 1

-

sin x sin (90 - lat.)   cos lat.   sec. lat.

1 __   1   _ 1

=

sin y   sin (90 - alt.)   cos alt.   sec. alt. Substituting these values in (114):

(      )

cost 2 0 = cos 2 (lat. + alt. + p.d.) cos lat. + alt. + p.d. - p.d.

X (sec. lat.) (sec. alt.)   (115)

The computation of the left-hand member can be simplified by expressing 0 in times of its supplement 0', that is, by reckoning azimuths from the south instead of from the north. Thus,

cost 2 0 = cost 2 (180° - YZX') = cost (90° - 2 0') = sine 2 0'

= 2 versin 0' = hav (180 - 0).

Half versines of angles, or " haversines " as they are called, are tabulated in Bowditch's American Practical Navigator and in other books on navigation.

99. The Directive Tendency of a Magnetic Compass. - A magnetic compass needle tends to set itself in the direction of the horizontal component of the magnetic field where it is situated. The magnetic poles of the earth do not coincide with the geographic poles. The magnetic north pole is situated in Boothia Peninsula, Canada, at latitude about 70° N., longitude about 964° W. - more than a thousand miles from the geographic north pole. The magnetic south pole is at latitude about 732° S., longitude about 1472° E. The number of degrees of angle between the geographic meridian at a particular place and the axis of a compass needle free to turn in a horizontal plane is called the magnetic declination at the particular place.

A line connecting all adjacent points on the earth at which the magnetic declination is zero is called an agonic line. One agonic