STELLAR PARALLAX. - The constant of parallax of a fixed star is the maximum angle which a line equal to the earth's mean distance from the sun would subtend if viewed at the star.

The distances of the fixed stars are so remote that till very recent times their parallaxes have been found to be insensible ; that is to say, the earth's orbit viewed from the nearest fixed star presents a disk (or ellipse) too small for measurement.

The limits of this article do not permit a detailed history of the early attempts of astronomers to determine the parallaxes of the fixed stars. The reader is referred on this point to Peters's 1Weis historique (les travaux $241. la parallaxe (les etoiles fixes, forming the first section of his celebrated work.Reekerehes sur la Parallaxe (les (Wiles Ares (Mem.. de l'Aca(I. Imp. de St .1)etersboury, sec. Math. et Physiques, vol. v.). The most notable incident in that history was the discovery- of aberration by Bradley, in 1728, when engaged in an unsuccessful attempt to determine the parallax of the star y Draconis.

The first determination of the parallax of a fixed star is' due to Henderson, ITis Majesty's astronomer at the Cape of Good Hope in 1832 and 18:33.3 It was followed by tlie nearly simultaneous discoveries of the parallax of 61 Cygni by Bessel 4 and that of a Lyrie by Struve 5 from observations made in the years preceding 1840. Since that time similar researches lrave been prosecuted with gradually increasing success.

The methods of observation rnay be divided into two classes, - the absolute and the differential.

The (tbRolute method depends on observation of the zenith distance of a star about the epochs of maximum parallactic displacement in declination - in practice, how-ever, generally throughout the whole year. The differences of declination so observed, after allowing for the effects of refraction, precession, aberration, nutation, and proper motion, afford the means of deducing the parallax of the star. The most notable series of observations of this charaeter are those of Maclear at the Cape of Good Hope, by which lie confirmed the results of his predecessor Henderson and those of Peters at Pulkowa in the second section of his work above mentioned. The latter is the most classic work in existence on refilled observations of absolute decli-nation, and it is by no means certain that, in more modern meridian observations, the work and methods of that dis-tinguished observer have been equalled - except perhaps at Pulkowa. The minute precautions necessary in such work will be found in Peters's paper above mentioned (see also TRANSIT CIRCLE). BLitOt with all the skill of Peters, nor with every refinement of equipment and obser-vation, can the difficulties caused by refraction and minute change of instrumental flexure, &c., be completely over-come; the method of absolute altitudes does not, in fact, respond in accuracy to the demands of the problem.

The diferenlial method depends on measuring the difference of declination, of distance, or of position angle between the star whose parallax is to be determined and one or more stars of comparison. It is assumed that the stars most likely to have sensible parallax are those which are remarkable for brilliancy or proper motion, and that the parallaxes of the stars of comparison (having little or no sensible proper motion and •faint magnitude) are so small as to be insignificant. So far as our know-ledge goes these assumptions are justified.

Researches on stellar parallax by these methods have been followed of late years with considerable success. The instruments employed. have been the heliometer and the filar micrometer (see MicuomETER, vol. xvi. pp. 243-248), the latter instrument being used in conjunction with an ordinary equatorial (see TELESCOPE). The precautions

The heliometer method seems to present the greatest facilities for extensive researches on stellar parallax, not only beca.use measures with this instrument seem, on tbe whole, to possess the highest accuracy, but because (on account of the large angles that can be measured) a much wider selection of suitable stars of comparison is availalule. Gyklen of Stockholm has applied the method of observing the differences of right ascension between the star whose parallax is to be determined and each of two comparison stars, and the same method has also been applied by Auwers (Math. Abhand. Berliner Acad., 1867); but the results obtained in this way do not compare at all favoura,bly with the accuracy of properly conducted helio-meter measu The diagram (fi.g-. 1) represents observations made by Gill to determine the parallax of a Centauri, with a helio-meter at the Cape of Good Hope. The ordinates of the curve are the time reckoned from 1882.0, the abscis&e the changes in the place of a Centauri due to the parallax . computed from the observations. Each dot represents the observations of each single night, and the reader will be able to judge of the accuracy of the observations from the a_PTeeinent of the dots with the curve. Fig. 2 in like manner represents a series of observations of Sirius.

These and many other results show that, with similar means, it is now 'possible to detect any differential parallax amounting to 0'05 with certainty, by a series consisting of a reasonable number of like observations - thus opening up a, wide and important field for future research.

The following table contains a list of those stars of which the parallax is known with considerable accuracy, - Nos. 1 to 13 being in the northern and Nos. 14 to 22 in the southern hemisphere.'

xxiii. p. 173 (0'260±0'020). 6, Krueger (heliometer), Ibid., (0"-24 7 ± 0"'021). 7. Brum-low, Du-nsink Observations, vol. ii.

p. 31 (0"'240±0".011). 8. O. Struve, Meni. Acad. St PetersA glance at the table is sufficient to show that neither ap-parent magnitude nor apparent motion alhuds a criterion of the parallax of any fixed star. Similar researches must, in fact, be carried out on a much more extended scale before any definite conclusions can be drawn. At present we can only conclude that different stars really diftr greatly in absolute brightness and absolute motion.

The following are the formuhe which will be found most useful in computing the corrections for parallax : - For the Hoon, and Planets.

Pat 71 - the equatorial horizontal parallax ; A = the distance of the object from the earth ; I C and C' = the geocentric and apparent zenith distances respec-tively; =the earth's radius corresponding to 0 ; a and a' = the geocentric and apparent right ascensions of the object respectively; =the hour angle of the object (reckoned + when west of meridian).

To find the parallax of the moon in zenith distance and azi-muth, from the observed (or apparent) zenith distance and azimuth. Put -y = (cp - 0') eos A'.

Then sin (('- C)= p sin 71- sin ((' - 7); siii (A/ p si rr sin (cp - 0') sin A, Tile corresponding quantities are found with all desirable precision for the sun and planets by the formulie - C' C= pr Sill - 7); or approximately-= 7r Sill C'; A' - A =pr. sin (cP - tir ) .A.' cos ; the latter quantity may generally be neg,leeted.

To find the parallax of the inoon in right ascension and decli-nation from the true (or geocentric) right ascension and declination.

d• • t p sin 7r cos ; Cos Put sin 0= then tan (a - a)= tan 0 tan (45°+ 0) tan t.

tan 7- tan cp' COs - a') COS[t a')] • e, p sin 7r sin O'cos (-y - 8) then tan (5 - 8') = tan 0' tan (45°+ 0') tan (7- 8).

To find the parallax of the moon in right ascension and docli-nation from the observed (or apparent) right ascension and deeli-nation.3 sin (a a') -p sin 7r cos o' sin t' tan 0' cos - tan 7 - - - - - , cos[t' - 1(a- a )] sill (8 8,) _ p Sill 71. ct,, sin (7 _ a,).

To find the parallax of the sun, planets, or comets in tight ascension or declination.4 a a, pa CoS Sill t' tan 7 = ; a a, _ sin (-y When the, distance of the ohjeet from the earth (A) is given (the earth's Mean distance from the sun being reekoned unity), i, usually the ease in ephemerides of minor planets and coinets, we have mean solar parallax 7r = The reader will find the proof of these formuhe in Chauvenet's Spherical and Praetical Astronomy, vol. i. pp. 104-127.

For the Parallax of the, Fixed Star.,'.

Put p = the maximum angle subtended by the mean distance of the earth from the sun at the distance of the star, =the star's annual parallax; E the obliquity of the ecliptic; C)and r - the sun's longitude and radius vector; I. To find the heliocentric parallax of a star in right ascension and declination, its annual parallax (p) being known.

ct' - = -pr see 5(eos C) sin a - sin C) COS E cos a); 5' - 8 = -pr sin acos E Sill 8 sin a - sin E COS 5) - pr cos 0 cos a.

As =i7r7n cos (0- M); AP =prm'cos j'D - M'); M. COS IN1 =sin a sin P +sin 8 cos a cos P ; 1 m'sin =s - (cos a cos P + sin 8 sin a sin P) cos E + COS sin P sin e]; cos )1.' = -1 [ sin a cos P - sin 5 cos asin Pl .

(D. Gl.)