the transits, will produce the requisite formulæ in a very simple manner. They are as follows:

cos of interval reduced to arc X cot dec. cot lat.

The first is for computing the latitude of the station when the declination of the star observed is known, or vice versa. The other two are for the purpose of determining at what altitude to set the instrument, and at what time to look for the transit of any given star. In the latter two an approximate value of the latitude is to be used.

For the description of a large transit instrument, contrived for rapid reversal in the prime vertical, and having the telescope at one end of the axis, so that the striding level need never be removed from the supporting axis, see Struve's account of the Pulkova instrument, in 468 of the Astronomische Nachrichten of Schumacher. See also the Washington Astronomical Observations of 1845, introduction p. li. and p. 131. In these instruments there are 16 wires, 7 on each side of the middle one. The transits are taken over seven, and the instrument being quickly reversed, the transits are taken over the same seven in the reverse order.

The following is an example of one of Struve's observations. The 2d column reads upward.

Time of meridian transit not corrected for azimuth error of the instrument,

A knowledge of the distance of each wire from the optical axis is unnecessary; for if this distance be denoted by e plus when the wire is N. of the optical axis, and

minus when it is south, we have, t and t' being the corresponding hour angles, & the declination, and the latitude,

If we make (t' + t) =s, and (t'—t) = u, we have

tan

tan cos 8 cos u

(1)

sin csin s sin u cos d sin

Formula (1) gives the declination, c being eliminated.

(2)

distance of the wire from the optical axis. The following is the application of (1) to the example above.

If the inclination of the axis be denoted by i, which is the mean of the two inclinations, telescope N. and telescope s., then + i should be used in place of in the above formulas, or the correction for the declination should be

φ

sin 28
dd=
sin 24

In the example above &=

Correction for inclination of axis dd =

Observed declination,

59° 11' 39' 071
0 .814

59° 11' 39 ⚫885

The declination thus found is exact only on the hypothesis that the azimuth of the axis

a

of rotation is zero. If there be an azimuth a we have r =

sin

for the angle at the pole

between the true meridian and the meridian of the instrument, or circle of declination perpendicular to the circle described by the optical axis of the instrument. The instant of transit of the star over the meridian of the instrument is the half sum of the times of corresponding transits E. and W. Thus for the star ● Draconis above, we have

The instant of meridian passage p requires a small correction for the difference of inclinations of the axis in the two verticals E. and W. Denoting the former by i, and the latter by i'

and hence the true time of meridian passage by the instrument instead of 184 48m 4109, as above, is p' 18 48 41.01.

= @

If denote the right ascension of the star, and e the error of the clock, let a': -e denote the time of passage of the star over the true meridian. Then, for the angle of the two meridians

T p' a' in time

and for the azimuth of the axis of rotation reckoned from the south round by the west,

With this the correction of the observed declination for the axis of rotation becomes

do (15) sin 1'' sin 28

For stars near the zenith may be used instead of 8, and the formula becomes

dd = (157)2 sin 1" sin 2

The clock error in the above example on 15th January, 184 48m wase = 8.31 The apparent right ascension of o Draconis a=

.. a' =

18 48m 50*17 18 48m 41.86

085 in time, and a = —

110 in arc

Finally the correction of the declination is

too small to notice.

d8000017

372

CONCLUDING NOTE.

The author had designed giving the theory of eclipses and occultations, with their application to the determination of longitude, and also Bessel's method of measuring an are of the meridian; but as longitude by moon culminations is the better mode, and as measurements of the meridian are not at present being carried on or immediately contemplated in this country, it was thought expedient not to increase the size of a work intended for very general use. As, however, the method by occultations requires only a common telescope and a good timepiece, the reader is referred to the Appendix of the Nautical Almanac for 1836, pp. 134 and 145, for the necessary formulas, and an example of the computation of longitude by an occultation. (See also Lee's Tables and Formulas, p. 78, Part III.)

TABLES.