Then, by (1), (2), (3),
61
Computation of Apparent Places.-In computing the Apparent Places of the several stars for the several dates of observation, I have used the formula,
Correction to Mean R.A. (in seconds of time)=Ee + Ff + Gg + Hh + L + 1−300 + qt where e, f, g, h, I are the constants for each star, given in the Nine-Year Catalogue; E, F, G, H, L the "Airy's Day Numbers," given for each day in the Nautical Almanac; g the star's Annual Proper Motion in Right Ascension; and the elapsed fraction of the year corresponding with the given date. The small correction for Daily Aberration has been neglected, a very precise knowledge of time being! as already stated, unnecessary,
Similarly, for Apparent North Polar Distance,
Correction to Mean N.P.D. (in seconds of are)=Ee' + Ff + Ggʻ + Hh' + L + l'-300 + q't
The Apparent Places thus found are those for Greenwich Mean Midnight of the several dates. The results, reduced to the several times of observation, are shown in Table II, further on, which exhibits the computations for latitude.
Routine of observations.-The instrument was erected on the 8th of January, and adjusted to the meridian the same evening. Observations for latitude were taken on the nights of January 9, 11, 12, 13, 15, 16, 20, 21, 23, and February 7. Clock and circumpolar stars were also observed on those nights, and the levelling of the cross-axis was carefully attended to. The times were taken with the Mean Solar Chronometer, ARNOLD and DENT, No. 1,207, kindly lent to me by Commodore CUMING, R.N., from H.M.S. Victor Emanuel. It has run pretty steadily. The use of a mean solar timekeeper for stellar observations introduces a great deal of unwelcome labour and complication into the reductions, but I had no alternative, as a chronometer rated to sidereal time could not be obtained.
The steadiness of the transit-instrument during frequent use and reversal is shown by the small change of the azimuth errors, which varied only from 6".39 W. (horizontal value) on the 9th January to 67.95 E. on the 7th February, the azimuth screws not having been touched after the first adjust- ment to the meridian; and it would appear from the observations that this change is mainly due to a slow progressive movement of the pier. Such steadiness of a portable instrument on a new and imperfectly isolated pier, extending over a period of 29 days, is very satisfactory.
Investigation for the latitude.--For the latitude reductions, the investigation is as follows :- Let Z and Z' be the true meridianal zenith-distances of the south and north stars respectively, d and d' their North declinations, and Z the latitude of the place; then,
L
d + Z Z
.". L · ↓ (ď + ď') + ↓ ( Z – Z')...........................................(1)
Now, let z, z' be the (supposed) observed zenith-distances of the south and north stars; n,s the readings of the north and south ends of the level for the south star, and w', s' those for the north star; the value of one division of the level, in seconds of are; r the correction for refraction for the south star, and that for the north star. And let it be assumed that the horizontal position of the level is that which corresponds to the condition, correction for level == 0.
Then, if (as in this case) the graduation of the level proceeds continuously from one extremity. towards the object end of the telescope, the central division being 40.00, and if p, p' be the reduc tions to the meridian in cases where the stars have been observed before or after the meridian passage, and ff' the corrections for flexure for either star, it follows that
C
b
[=== }; ( d + d′ ) + § (m' — m) M + = {(w'−n) + (8'−s)} +4 (r−r') + § (p' − p) + 4 (ƒ −ƒ' )..............(4)
4
This is the final expression for the latitude. As already mentioned, M=68′′698, and'-1.067. The correction to the meridian (p or p') may be computed by the equation,
2 sin' i cos L cos d p
sin 1"
X
sin Z
where i is the star's hour angle from the meridian, east or west; but it has not been necessary to use it in this case, all of the stars having been observed on the meridian, or so nearly so as not to necesitate any correction on that account. The correction for flexure, being only that which is due to the difference of the zenith distances of the two stars, is also inappreciably small, and has been neglected.
The correction for refraction, though small, is appreciable, and has been applied. It is found by the equation,
or, by (3),
(r-r)= {57′′7sin (2-2′) sec *2}
↓ (r) = (57.7sin (m' ~m) M sec '*}
***
(5)
This correction bears the same sign as the quantity m'-m, as appears from this equation, as well as from the consideration that r-r'is positive only when z is greater than 2', in which case, under the given conditions, m'-m is also positive.
Tabulation of results.-The observations, reductions and results will be found embodied in the following Table, wherein Column I gives the date of observation; Column II the star's name, as in Table I; Column III its B.A.C. number; Column IV its magnitude, from the Greenwich Nine-Year Catalogue for 1872; Column V its position, North or South of the zenith; Column VI the position of the instrument, Circle East or Circle West; Column VII the star's Apparent Right Ascension at transit over the meridian of the place; Columns VIII, IX the chronometer times of bisection and culmination, to the nearest second; Column X the interval, in chronometer time, between bisection and culmination; Column XI the approximate error of the chronometer for each date; Column XII the micrometer readings, m,m', for the two stars of each pair, in revolutions and parts; Column XIII the quantity (mm); Column XIV, XV the readings of the north and south ends of the level, for both stars of each pair; Column XVI half the difference of the level indications for the two stars, or the quantity {(n'~n) + (s'~s)}; Column XVII the star's Apparent Declination at transit over the meridian; Column XVIII half the sum of the declinations for each pair; Columns XIX, XX, XXI the corrections for micrometer, level and refraction, computed as already explained [see equations (4) and (5)]; and Column XXII the resulting value of the latitude from each pair.
Z
-2(n
3(n + 8
-80) +r − p +ƒ
b
whence,
-
b
Z' = 2' — — ¡ (n' + s′ — 80 ) + r' —p'′ +f'
2
be
↓ (2− Z ) = ↓ ( z − 2') + ~ { (n' ~ n) + (8' − 8 ) } + ↓ ( r' − r') + ¿ (p' − p) + } ( ƒ −ƒ'................( 2 ) In the case of the zenith-telescope, z, z are not measured directly, but only their difference, z-3, by means of the micrometer. To find the expression for this in terms of the micrometer; let o the zenith distance corresponding to the central position of the micrometer thread; m, n the micro- meter readings for the south and north stars, in revolutions of the micrometer screw; and M the value, in seconds of are, of one revolution of the screw. Then, for that position of the instrument in which increase of micrometer readings corresponds with decrease of zenith distance, it is plain that
22
-m M
• m'M
(3-2)
(m' ~ m) M
(3)
In practice, to avoid confusion of signe, this position was always adhered to. It corresponds with the precept to observe stars. North of the zenith with Circle East, and those South of the zenith with Circle West.