correction is required on account of the diurnal aberration of a star in right ascension. (art. 237.)

292. Besides the deviations of the apparent from the mean places of stars, as above indicated, many of the stars have apparent motions which cannot be referred to the causes by which these deviations are produced; and hence they are supposed to be peculiar to the stars themselves, either independently of the other bodies of the universe or results of some general movement, as yet unknown, of the whole system of stars. In the Memoirs of the Astronomical Society, vol. v., Mr. Baily has given a list of 314 stars which are supposed to have proper motions; and in a few cases the directions and annual amounts of these motions have been ascertained. They are represented in the formulæ of reduction above mentioned by Ac for right ascension and by ▲ c' for declination; and the values of these terms are to be multiplied by t which expresses the fraction of a year elapsed days since Jan. 1. between Jan. 1. and the given instant, or t = 365.25

The star a Centauri is one of those whose proper motion is the greatest; and the amount of the motion annually is said to be equal to 3"-6 of a degree.

293. The stars called fixed have at all times been classed according to certain orders of apparent magnitude or brightness, from the most brilliant, which are said to be of the first order; the stars of the seventh order being those which are barely visible to the unassisted eye. Stars of the same apparent class are not however equally brilliant; and Sirius is supposed to emit about three times as much light as an average star of the first order of magnitude. Many of the stars also suffer periodical changes in their degrees of brilliancy, some appearing constantly to diminish, and others to increase in brightness: a few which are mentioned in the ancient catalogues are not now to be found in the heavens, and some are there at present which do not appear to have been formerly remarked. It may be added that some of the nebulous spots are known to have changed their figure since they were first noticed.

In order to form a judgment of the relative magnitudes of stars from the first to the sixth or seventh order, and even to estimate any changes which may take place in the apparent magnitudes of stars within those orders, some astronomers trust to the impression produced by their different brilliancies on the unassisted eye: a greater degree of precision in the estimate may, however, be obtained by comparing a star which may be under notice with one of a known order,

in the telescope of a sextant, after having by a movement of the index brought the two stars together in its field. The relative quantities of light received from stars of different magnitudes can be ascertained with considerable precision by diminishing the aperture of a telescope (the object glass if the telescope is achromatic, or the speculum if a reflector) when viewing the brighter of two stars, till the eye is affected in the same manner as it is when the less bright is viewed with the entire aperture; for the quantities of light emitted from the two stars will evidently be inversely as the areas or as the squares of the diameters of the apertures with which the intensities of light appear to be equal. By such experiments Sir John Herschel concludes that the light of an average star of the first magnitude is to that of a star of the sixth as 100 to 1. (Mem. Astr. Soc. vol. iii. p. 182.)

In estimating the relative magnitudes of stars below the sixth the same astronomer proceeds on the principle that the designation of magnitude should increase in an arithmetical progression 7, 8, 9, &c., while the light decreases in the geometrical progression,,, &c.; so that if two stars apparently of the same magnitude, when brought together so as to appear but as one, affect the eye in the same manner as one of the sixth magnitude, each is to be considered as of the seventh magnitude, and so on. The sensibility of the eye to differences in the degrees of illumination is very great; and, with telescopes of considerable power, the organ is capable of appreciating magnitudes of the 18th or 20th order.

It is customary to express degrees of magnitude between the primary orders by whole numbers with decimals; thus mag. 3.5 denotes an order equally distant between the third and fourth.

294. Mayer seems to have been the first who observed that several of the large stars are accompanied by smaller ones which revolve about them like satellites: such have since been discovered in almost every part of the heavens; and Sir John Herschel has given, in the Memoirs of the Astronomical Society, a table of the positions of 735 double stars in the northern hemisphere, to which must now be added those which the same astronomer observed in the southern hemisphere during his residence at the Cape of Good Hope. One of the largest double stars in our hemisphere is a Geminorum (Castor) of which the small one is said to have described about the other, between the years 1760 and 1830, an angle of about 67 degrees, with a variable velocity. The star & Ursa Majoris is double, and the smaller one re

volves about the other in a period of about 59 years, which was completed between 1781 and 1839. One of the two stars constituting Coronæ is supposed to have completed its period in 45 years; and from certain variations in its angular motion (if the observations can be depended on) it may be presumed that its orbit is elliptical. The stars e1 and 2 Lyræ, from a similarity in appearance, and from an equality of angular motion, seem to constitute a binary system, the two stars having a combined rotation, in the same direction, about their common centre of gravity.

295. The angular distances between two stars, one of which revolves about the other, and also their angle of position with respect to the meridian or to a horary circle passing through one of them, when measured with a micrometer, is liable to much uncertainty from the imperfection of the instrument and the difficulty of adjusting it with sufficient precision for the purpose; so that the path which the revolving star describes about that which is fixed, if determined directly from the measures, would appear to have no regular figure. An ingenious method, however, of obtaining by a graphical construction, from the observations, an approximation to the true values of those distances and angles, and subsequently of ascertaining nearly the figure of the path or orbit, has been given by Sir John Herschel in the fifth volume of the Memoirs of the Astronomical Society.

This consists in laying down, by any scale of equal parts, on a line taken at pleasure a series of distances representing the intervals (in years and decimals) between the observations; and from these points setting up, at right angles to the line, ordinates whose lengths may represent the observed angles of position: then a line being drawn by hand among the extremities of the ordinates so as to present the appearance of a regular curve of gentle convexity or concavity, its ordinates may be conceived to represent with tolerable correctness the values of the angles of position for the times indicated by the corresponding abscissæ on the line first laid down. Next, drawing tangents of any convenient length to this curve, at different points in it, and considering them as the hypotenuses of right-angled triangles whose bases are parallel to the line of abscissæ, and represent time, while the perpendiculars represent angular motions in the orbit; the quotients of the latter by the former denote the angular velocities of the star in the orbit, at the times indicated on the scale of abscissæ, corresponding to the points of contact. Now it being presumed that the path really described by one star about another is a circle or an ellipse, and the ap

parent path in the heavens being an orthographical projection of the orbit on a plane perpendicular to the line drawn from the spectator to the star, the latter path will (by conic sections, and art. 27.) also be an ellipse: hence, by the elliptical theory for planets (art. 259.), the squares of the radii vectores of the revolving star (the other being at the focus) will be inversely proportional to the angular velocities; therefore, by drawing lines from the star, supposed to be at the focus, making angles with each other equal to the velocities just found, and making the lengths of the lines inversely proportional to the square roots of those velocities, the curve joining the extremities, or passing among them, may, as in the method of finding the orbit of a planet, be expected to approach nearly to the form of an ellipse. Let the curve so drawn be an ellipse, as AZBX; let c be its centre and s be the place of the star about which the other star revolves: then in order to form a scale of seconds of a degree for the diagram, take from the scale by which the projected figure was drawn a mean of all the radii, or lines drawn from s to

A'

n

M

K

Z

B

C/

B

P R

A

X

L

N

T

the periphery, for which the corresponding distances in seconds have been obtained by measurement with the micrometer. From a mean of all these measured distances compared with the mean radius there may be found, by proportion, the number of seconds corresponding to each unit of the scale employed in the projection.

296. Since by the nature of the orthographical projection the centre of the projected ellipse corresponds to the centre of the real ellipse, if a line, as AB, be drawn through s and C, that line will be the projection of the major axis, and sc will be that of the excentricity; also by the nature of the projection, these will bear the same ratio to one another as the real major axis and excentricity: therefore we have for the value of e (the excentricity when the semiaxis major unity). Let a' be the real semi-axis major, b' the real semiaxis minor: then e, and the ratio of a' to b' is known.

a'

b'

SC

AC

is

Next, by trials find the position of a line KL drawn through s so as to make the parts SK, SL equal to one another; then KL will be the projected latus rectum (for in the real orbit,

a double ordinate equal to the latus rectum would pass through s and be bisected in that point), and consequently zx drawn through c parallel to KL is the projected minor axis, hence measuring AB and ZX by the scale, the ratio (= %) is

AB

ZX

a

known (a and b being the projected semiaxes). Let the line ns passing through the star s be the direction of the meridian, or the hour circle, from which the angles of position in the heavens were measured; and let it cut zx in R; then with a protractor or otherwise measure the angles nSB, nSK (= nRZ), or the inclinations of the projected axes to that line: let these inclinations be represented by a and ẞ respectively.

Again, let A'Z'B'X' be the real orbit, of which the other is presumed to be an orthographical projection formed by lines, as A'A, B'B, c'c, &c. drawn from every part of the real orbit perpendicularly on the plane of the apparent orbit, and let MN be the line of section; then AB, ZX being the projected axes, A'B' and z'x' will be the corresponding axes of the real orbit. Let the planes z'z XX', ZMNX be produced till they meet the plane A'X'B' in some point as T, and let the angle nsм be represented by n: then the angle STC (: =MSK) = ß — n, and the exterior angle MSC = a — CST=(an). Now imagine CP to be let fall perpendicularly on ST, and c', P to be joined: then (Geom. 2 and 4. Def. Planes) the angle CPC (7) is the inclination of the projected to the real orbit.

C

-n, or

Next, imagine c' to be the centre and cc' the semidiameter of a sphere in contact with the plane SCT at the point c; then the plane angles Sc'c and TC'c may be represented by two sides of a spherical triangle intersecting one another at C, and SCT as the spherical angle continued between them. Let acb, in the annexed figure, be such a triangle, and from c let fall the perpendicular cd; then cd will represent the angle cc'P in a plane passing through cc' perpendicularly to the base ST of the plane figure,

S

a

d

P

T

that is, it will represent the complement of the angle CPC' which is the inclination of the plane sc'T to SCT.

Now, in the spherical triangle acd, right angled at d, we have (art. 60. (d)) cos. ac = cos. ad cos. C d,

and substituting the equivalents,

cos. CC's cos. SC'P cos. CC'P, or sin. CSC'sin. C'SP sin.C PC':