The substance of the above rule may be otherwise stated as follows: The astronomical time is the hours and minutes elapsed since the NOON LAST PAST, the astronomical DATE being that of the civil day to which the noon belongs. Thus, April 23, 4.15 p. m., civil time, is April 23, 4h. 15m., astronomical time, and April 23, 4.15 a. m., civil time, is April 22, 16h. 15m., astronomical time.

Hour Angle of Polaris.-In Fig. 9 the full vertical line represents a portion of the meridian passing through the zenith Z (the point directly overhead) and intersecting the northern horizon at the north point N, from which, for surveying purposes, the azimuths of Polaris are reckoned east or west. The meridian is pointed out by the plumb-line when it is in the same plane with the eye of the observer and Polaris on the meridian, and a visual representation is also seen in the vertical wire of the transit, when it bisects the star on the meridian.

When Polaris crosses the meridian it is said to culminate; above the pole (at S) the passage is called the upper culmination, in contradistinction to the lower culmination (at S').

In the diagram-which the surveyor may better understand by holding it up perpendicular to the line of sight when he looks toward the pole Polaris is supposed to be on the meridian, where it will be about noon on April 10 of each year. The star appears to revolve around the pole, in the direction of the arrows, once in every 23h. 56.1m. (23 hours, 56 minutes, 4.09 seconds) of mean solar time; it consequently comes to and crosses the meridian, or culminates, nearly four minutes earlier each successive day. The apparent motion of the star being uniform, one quarter of the circle will (omitting fractions) be described in 5h. 59m., one-half in 11h. 58m., and threequarters in 17h. 57m. For the positions 81, 82, 83, etc., the angles SPS1, SPS2 SPs, etc., are called Hour Angles of Polaris for the instant the star is at S1, S2, or 83, etc., and they are measured by the ares SS1, SS2, Ss3, etc., expressed (in these instructions) in mean solar (common clock) time, and are always counted from the upper meridian (at S) to the west, around the circle from Oh. Om. to 23h. 56m.1, and may have any value between the limits named. The hour angles,

measured by the arcs S8, S8, S8, S8, S8, and Sse, are approximately 1h. 8m., 5h. 55m., 9h. 4m., 14h. 52m., 18h. 01m.,' and 22h.

48m. respectively; their extent is also indicated in the diagram by broken fractional circles about the pole.

The hour angles, 5 h. 55 m. and 18 h. 01 m., are those at west and east elongation respectively in latitude 40° N.

Hence, to obtain the hour angle of Polaris subtract the time of upper culmination from the correct local mean time of observation; the remainder will be the hour angle of Polaris.

The observation will be made as directed under Method I, modified as follows: there will be no waiting for the star to reach elongation; the observation may be made at any instant when Polaris is visible, the exact time being carefully noted.

TABLE XVII.

This table gives, for various hour angles, expressed in mean solar time, and for even degrees of latitude from 36 to 40 degrees, the azimuths of Polaris during the remainder of this century, computed for average values of the north polar distance of the star-the arguments (reference numbers), being the hour angle (or 23h. 56m.1, minus the hour angle, when the latter exceeds 11h. 58m.), which is termed the Time Argument;' and the latitude of the place of observation. The table is so extended that azimuths may be taken out by mere inspection and all interpolation avoided, except such as can be performed mentally.

The hours of the "time arguments are placed in the columns headed "Hours," on the left. The minutes of the time arguments will be found in the columns marked " m.," under the years for which they are computed, and they are included between the same heavy zigzag lines which inclose the hours to which they belong.

The vertical diameter SS', Fig. 9, divides the apparent path of Polaris into two equal parts, and for the star at any point s, on the east side, there is a corresponding point s on the west side of the meridian, for which the azimuth Nw is equal to the azimuth Ne. The arc Ss,S's., taken from the entire circle (or 23 h. 56.1 m.), leaves the arc Ss,, and its equal, Ss1, expressed in time, may be used to find, from Table XVII, the azimuth Nw, which is equal to Ne.

The hour angles entered in Table XVII include only those of the west half of the circle ending at S', and when an hour angle greater than 11 h. 58 m. results from observation, it will be subtracted from 23 h. 56.1 m., and the remainder will be used as the "time argument" for the table. The surveyor should not confound these two quantities. The hour angle itself always decides the direction of the azimuth and defines the place of the star with reference to the pole and meridian, as noted at top of Table XVII. See examples at the end of this part.

The time arguments are given to the nearest half-minute; the occurrence of a period after the minutes of any of them indicates that its value is 0.5m. greater than printed, the table being so arranged to economize space.

The table will be used as follows: Find the HOURS of the time argument in the column marked "Hours"; then, between the heary lines which inclose the hours, find the MINUTES in the column marked at the top with the current year. On the same horizontal line with the MINUTES the azimuth will be found under the given latitude, which is marked at the top. Thus, for 1897, time argument, Oh. 41m., latitude 38°, is the azimuth 0° 17'. For 1899, time argument 7h. 53m., latitude 36°, the azimuth is 1° 19′.

If the exact time argument is not found in the table the azimuth should be proportioned to the difference between the given and tabular values of said argument. Thus, if the time argument in the first of the above examples (for 1897) was Oh. 39m., instead of Oh. 41m., the azimuth would be the mean between 0° 15′ and 0° 17′ or 0° 16'. In a similar manner, if the latitude is nearer an odd than an even degree, the mean of the azimuths for the next greater and next less latitude will be used; thus in the above example for 1899, if the given latitude was 37°, the mean between 1° 19' and 1° 21', or 1° 20′, would be corresponding azimuth. The table has been arranged to give the azimuths, as exemplified above, by simple inspection. No written arithmetical work is required, all being performed mentally. It will generally be sufficient to take the nearest whole degree of latitude and use it as above directed; for a few values near the bottom of the table, for example, the latitude may be taken to the nearest half degree.

The attention of the surveyor is directed to the fact that he should always use one day of twenty-four hours as the unit when he subtracts the time of culmination from the time of observation. In any case when the time of upper culmination taken from Table XV, for the given date, would be numerically greater than the astronomical time of observation, the former time will be taken out for a date one day earlier than the date of observation. The surveyor will decide

when such conditions exist by comparing the time given in the table with his astronomical time of observation. The upper culmination to be used at any time will always be the LAST one that occurs before the observation.

When an hour angle comes out within one minute of either Oh. Om., or 23h. 56m.1, the observation may be regarded as having been taken with the star on the meridian, above the pole; if within one minute of 11h. 58m., Polaris may be considered on the meridian below the pole at the time of observation.

At elongation Polaris is nearly 5h. 55m. west (or east) of its position at upper culmination; consequently if the hour angle for any observation comes within five minutes of 5h. 55m., or 18h. 1m., the star may be assumed to be at elongation, west for the first and east for the second hour angle, and its azimuth may be taken from a preceding table (No. XVI), which gives its value at elongation from 1890 to 1910 inclusive.

Should the surveyor wish the time of lower culmination, for use with the plumb-line method (No. II), described on page 516, or for any other purpose, he will first determine the time of upper culmination for the date (Table XV) and then subtract 11h. 58m. for the preceding lower culmination, or add 11h. 58m. for the lower culmination following the derived time for upper culmination, attending to the addition or subtraction of 23h. 56.1m., as directed in an example (1) below.

The time to be used when making observations on Polaris off the meridian should be as accurate as can be obtained. Looking at Table XVII, near the top of the page, the surveyor will observe that for a difference of four minutes in the time argument there is a change of about two minutes in azimuth; consequently, to obtain the azimuth to the nearest whole minute of arc, the local mean time, upon which all depends, should be known within two minutes. When the surveyor uses a solar instrument, he can readily determine the time for himself during the afternoon before observing Polaris, or in the morning after observation, and, without moving the hands of his watch, apply the necessary correction to his observed watch time. When