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log sin (to — t) = 9-5732 log sec (5 — c) = 0.0044 log sin (<t> - 5) = 9-7373 log sec <f> = 0.1313 log 4 s= 0.6021

* The factor 4 has been introduced in the following example in order to reduce minutes of angle to seconds of time.

The true sidereal time may now be found by subtracting 1» 20^.3 from the right ascension of o Virginis, the result being as follows:

a = oo"» 26s.$ t = —1 20.3 S = n* 59m oós.o

The local civil time corresponding to this instant of sidereal time for the date is 2o& 55»» 14^.5. tThe corresponding Eastern Standard time is 20^ 39™ 32^.8, or Sh $gm 3 2s.8 p.m. The difference between this and the watch time, 8*39"» 43*, shows that the watch was 10s.2 fast.

84. Time by Equal Altitudes of a Star.

If the altitude of a star is observed when it is east of the meridian at a certain altitude, and the same altitude of the same star again observed when the star is west of the meridian, then the mean of the two observed times is the watch reading for the instant of transit of the star. It is not necessary to know the actual value of the altitude employed, but it is essential that the two altitudes should be equal. The disadvantage of the method is that the interval between the two observations is inconveniently long.

85. Time by Two Stars at Equal Altitudes.

In this method the sidereal time is determined by observing when two stars have equal altitudes, one star being east of the meridian and the other west. If the two stars have the same- declination then the mean of the two right ascensions is the sidereal time at the instant the two stars have the same altitude. As it is not practicable to find pairs of stars having exactly the same declination it is necessary to choose pairs whose declinations differ as little as possible and to introduce a correction for the effect of this difference upon the sidereal time. It is not possible to observe both stars directly with a transit at the instant when their altitudes are equal; it is necessary, therefore, to observe first one star at a certain altitude and to note the time, and then to observe the other star at the same altitude and again note the time. The advantage of this method is that the actual value of the altitude is not used in the computations; any errors in the altitude due either to lack of adjustment of the transit or to abnormal refraction are therefore eliminated from the result, provided the two altitudes are made equal. In preparing to make the observations it is well to compute beforehand the approximate time of equal altitudes and to observe the first star two or three minutes before the computed time. In this way the interval between the observations may be kept conveniently small. It is immaterial whether the east star is observed first or the west star first, provided the proper change is made in the computation. If one star is faint it is well to observe the bright one first; the faint star may then be more easily found by knowing the time at which it should pass the horizontal cross hair. The interval by which the second observation follows the time of equal altitudes is nearly the same as the interval between the first observation and the time of equal altitudes. It is evident that in the application of this method the observer must be able to identify the stars he is to observe. A star map is of great assistance in making these observations.

The observation is made by setting the horizontal cross hair a little above the easterly star 2» or 3W before the time of equal altitudes, and noting the instant when the star passes the horizontal cross hair. Before the star crosses the hair the clamp to the horizontal axis should be set firmly, and the plate bubble which is perpendicular to the horizontal axis should be centred. When the first observation has been made and recorded the telescope is then turned toward the westerly star, care being taken not to alter the inclination of the telescope, and the time when the star passes the horizontal cross hair is observed and recorded. It is well to note the altitude, but this is not ordinarily used in making the reduction. If the time of equal altitudes is not known, then both stars should be bright ones that are easily found in the telescope. The observer may measure an approximate altitude of first one and then the other, until they are at so nearly the same < altitude that both can be brought into the field without changing the inclination of the telescope. The altitude of the east star may then be observed at once and the observation on the west star will follow by only a few minutes. If it is desired to observe the west star first, it must be observed at an altitude which is greater than when the east star is observed first. In this case the cross hair is set a little below the star.

In Fig. 62 let nesw represent the horizon, Z the zenith, P the pole, Se the easterly star, and Sw the westerly star. n * Let te and tw be the hour angle of Se and Sw, and let HSeSw be an almucantar, or circle of equal altitudes.

From Equa. [37], for the two stars Se and Sw, the sidereal time is

Taking the mean value of

Fig. 62

Fig. 62

from which it is seen that the true sidereal time equals the mean right ascension corrected by half the difference in the hour angles. To derive the equation for correct ing the mean right ascension so as to obtain the true sidereal time let the fundamental equation sin h = sin 8 sin <f> + cos 8 cos <f> cos t [8]

* te is here taken as the actual value of the hour angle east of the meridian.

be differentiated regarding 6 and t as the only variables, then there results

• dt o — sin <t> cos d — cos 6 cos sin t-^ — cos <j> cos t sin 5, [gQ]

from which may be obtained dt _ tan <j> __ tan 8

If the difference in the declination is small, db may be replaced by h(8w — 5«), in which case dt will be the resulting change in the hour angle, or J (/«, — te). The equation for the sidereal time then becomes a atw + oie . 8w — 8e f~tan 4> tan r i

in which (8w — 5«) must be expressed in seconds of time. 8 may be taken as the mean of 8e and 5». The value of t would be the mean of te and tw if the two stars were observed at the same instant, but since there is an appreciable interval between the two times t must be found by ae — aw , Tw — Te , ,

Tw and Te being the actual watch readings.

LIST FOR OBSERVING BY EQUAL ALTITUDES Lat., 42o 21' N. Long., 4h 44"» 18® W. Date, Apr. 30, 1912.

Stars.

Magn.

Sidereal time of equal altitudes.

Eastern time of equal altitudes.

a Corona Borealis

f Geminorum

a Bootis

5 Geminorum

p Bootis

p Argus

0 Herculis

a Serpentis

a Canis Minoris.

0 Herculis

S Geminorum.

a Serpentis

0 Cancri

a Serpentis

Hercuhs

y Cancri

0 0

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