In the triangle SA'T we have (art. 61.)

whence

sin. ST sin. A':: sin A'T: sin. A'ST;
sin. A' sin. A'T

sin. ST

sin. A'ST = cos. ss't.

And (reckoning in the order A'OET, that is of the signs) sin. A'T sin. (Y T Υ Α') sin. Y T cos. Y A'

-cos. T sin. YA'; (Pl. Trigon., art. 32.) therefore sin. A' sin. A'T = sin. A' cos. Y A' sin. YT

... ·

— sin. A' sin. YA' cos. Y T (A). Now in the triangle PA'B, right angled at B (art. 62. (e')),

r sin. A'B cotan. A' tan. PB

But B270°, according to the order of signs; therefore A'B = 270°. YA', and sin. A'B-cos. A': substituting this term for its equivalent, and multiplying both members by sin. A', the last equation becomes, cos. A' being negative,

r sin. A' cos. YA' cos. A' tan. PB. Also, in the same triangle we have (art. 60. (ƒ)), r cos. Asin A'PB cos. PB;

therefore the preceding equation becomes

sin. A' cos. Y A' sin. A'PB sin. PB.

But A'PB 90° + M; therefore sin. A'PB = cos. Y M = cos. a (a representing the star's right ascension): also PB = 90° + Pp; therefore

sin. PB

cos. Pp cos. 0 (0 representing the obliquity of the ecliptic).

Consequently sin. A' cos. Y A' cos. a cos. e.

Again, in the triangle Y A'M, right angled at M (art. 60. (e)), sin. A′ sin. Y A′ = sin. Y M = sin. a.

therefore

sin. T=

Lastly, T (according to the order of signs) = L+ 270°; cos. L, and cos. Y T = sin. L. Then, substituting all the values just found in the equation (A) we get sin. A'sin. A'T=-(cos. L cos. a cos. + sin. L sin. a). Now s't (= ss' cos. ss't) = 20′′.36 sin. ST cos. SS'T,

or

Hence it follows that

s't=

20.36 sin. A' sin. A'T:

20.36 (cos. L cos. a cos. + sin. L sin. a), which being divided by cos. SM, or cos. d (d representing the star's declination) is reduced to the equator, and becomes the value of MN, the aberration in right ascension.

235. To find st the aberration in declination.

Imagine the great circle s'I', of which s't may be considered as a part, to be drawn through s' perpendicularly to PS, cutting the equator in w, and the ecliptic in I': then w will be the pole of PSM, and, st being very small, the angle at w may be considered as measured by Ms, the star's declination.

Now in the triangle s'I'T (art. 61.)

(:

sin. S'T: sin. I′ :: sin. I'T: sin. I's'T (= sin. ss't);

therefore, putting ST for S'T, sin. I′s ́T =

sin. I' sin. I'T

sin. ST

But, reckoning according to the order of the signs, I′T=

therefore (Pl. Trigon., art. 32.) sin. I'T sin. Y T cos. Y I'cos. Y T sin. TI',

and sin. I' sin. I'T sin. I' cos. I' sin. T

Now in the triangle I'w

therefore

I′ cos. Y T. (B).

(art. 61.)

sin. I' sin. I' sin. W sin. WY:

but (reckoning according to the order of the signs) Mw=270°; therefore w = √ м + 270° = a + 270°; and sin. y w = cos. a, also cos. Y w sin. a.

Again, multiplying both members of the last equation by cos. I'v

sin. I'

or by cotan. I', that equation becomes

sin. I' cos. I'y sin. w sin. W Y cotan. I'√ : but (art. 62., 2 Cor.) sin. w cotan. I'=cos. wr cos. r + cotan. I'w sin. Y;

also, as above, cos. Y T sin. L, and sin. YT =

COS. L;

therefore substituting the values of the terms in the above equation (B) we have

sin. I' sin. I'T cos. L sin. d sin. a cos. + cos. L cos. d sin. 0 sin. L sin. d cos. a;

which being divided by sin. ST in order to have the value of sin. I's'T, or ss't, and then multiplied by ss', or by its equivalent, 20".36 sin. ST, the result will be the value of st, the required aberration in declination.

236. The value of the angle EME' (fig. to art. 229.) for any star, being supposed to be determined by observation, and that of the angle SE'T being obtained by computation; it will follow, by trigonometry, that the ratio of ME' to EE' is

nearly as 10,110 to 1; and since it is not convenient, in this place, to introduce an account of the celestial observations by which the velocity of light was first determined, it will be proper at present to admit the motion of light as an hypothesis, and to consider the ratio above mentioned as a result of the observations by which the aberration was ascertained. It will then follow, the distance of the earth from the sun and the mean daily motion of the former in its orbit being supposed to be known, that the earth must describe in the orbit an arc subtending at the sun an angle of 20".36 in the time (8′ 13′′.5, art. 282.) that a particle of light passes from the sun to the earth; and hence it will be evident that when the sun's centre appears in the axis of a telescope, that centre is apparently 20".36 in advance of the place which it would occupy if no time elapsed in that passage. The effect of aberration on the place of the sun is, therefore, the same as that which takes place on a star situated in the pole of the ecliptic. Aberration necessarily affects also the apparent places of the planets, and its value for each depends upon the relative motions of the light, the earth and the planet. The formule by which its effects are computed are investigated in several treatises of astronomy. lambre," Astronomie," tom. iii. ch. 29.

See De

237. The diurnal aberration of light is a phenomenon resulting from the movement of light combined with the rotation of the earth on its axis: and it differs from the annual aberration above described, in consequence merely of the difference between the velocity of the earth on its axis and in its orbit: now the annual movement of the earth in the time in which light passes from the sun to the earth is, as above mentioned, 20".36; but if r be the semidiameter of the earth, and that of the earth's orbit supposed to be circular, we have = (p being the sun's equatorial

parallax, or 8′′.6);

r

sin. p therefore the movement of the earth in its

orbit in that time is

r

sin. p

20".36 sin. 1", or putting 8".6

sin. 1′′ for sin. p, the movement in the orbit is

20".36 r,

8".6

or

2.355 r. The angular motion of the earth on its axis in the same time is equal to 2° 3′ 18′′, or in linear measure, 7398" sin. 1′′. r; and the annual aberration of a fixed star when a maximum is 20".36: therefore

2.355r7398" sin. 1" :: 20".36

0.3085,

or in sidereal time, 0" 0206. This is the diurnal aberration,

in right ascension for a star in the equator at the time that it is on the meridian of a station, the latter being on the terrestrial equator; but for a station whose latitude is L, the diurnal rotation being less on the parallel of latitude than at the equator in the ratio of radius to cos. L, the aberration of a star on the meridian of the station is, in sidereal time, equal to 0.0206 cos. L; if the star be not in the equator, this expression denotes only the aberration on the parallel of the star's declination, and in order to reduce it to the celestial equator, or to the value of the aberration in right ascension, it must be divided by the cosine of the star's declination. The diurnal aberration in declination is too small to require notice.

238. The displacement of the pole of the world in consequence of the general retrogradation of the equinoctial point is such that, while the latter point would appear to describe the circumference of the ecliptic in the heavens, the pole of the equator would describe a circle about the pole of the ecliptic with an equal angular motion (50′′.2 annually), and at a distance equal to the mean obliquity of the two circles. But Dr. Bradley remarked that the observed declinations of stars differed from those which resulted, by computation, from the hypothesis of a uniform precession by small quantities which varied very slowly between the extreme limits (about + 9" and 9"), and he discovered that the star returned to the same position in declination in about eighteen or nineteen years; in which time either of the nodes of the moon's orbit revolves about the earth. It may be observed that this apparent displacement of the stars was a discovery made subsequently to that of aberration; and that its effects in the course of one year are much smaller than those which result from the movement of light and of the earth.

SHI

N

с

B

D

L

S"

239. In order to explain the nature of the displacement let QN be the mean place of the celestial equator and p its pole at a time when the moon's ascending node is at the equinoctial point Q. At such time it was observed by Bradley that the polar distance of a star s whose right ascension was six hours, instead of being represented by PS, was less than the value of that arc by about 9"; while the polar distance of a star s' whose right ascension was eighteen hours exceeded PS' by the same quantity; at the same time a star at s" or s", whose right ascension was O or twelve hours, appeared to suffer no change of declination; and therefore the phenomena were

N/

such as would take place if the pole of the equator were at A, at a distance from P equal to about 9" of a degree. Again, let QN represent the mean place of the equator and p its pole at a time when, by its retrogradation, the moon's node is at N, its longitude being then 270 degrees. At such time (about four and a half years from the former time) the polar distance of a star s'" whose right ascension is zero, instead of being represented by PS"" was less than the value of that arc by about 7′′, while that of a star s" whose right ascension was twelve hours exceeded PS" by the same quantity: at that time a star at s or s' appeared to suffer no change of declination; therefore the phenomena were such as would take place if the pole of the equator were at B, at a distance from P equal to about seven seconds. When the moon's node was at L (about nine years from the commencement of the observations) the observed declinations of stars were such as would take place if the pole of the equator were at C in the solsticial colure NPN', at a distance from P equal to AP; and during the following nine years the variations of declination were observed to take place as in the preceding period but in a reverse order. These observations were made between the years 1727 and 1746, and Dr. Bradley's conclusion was that the phenomena might be represented by supposing an oscillation of the plane of the equator upon the line of the equinoxes, combined with the general movement of that line on the centre of the ecliptic; so that the pole of the world, or the extremity of the axis of the equator, which axis is always at right angles to the plane of that circle, seems to describe a small elliptical figure ABCD about the point P, the mean place of the pole.

240. It is easy to conceive, since P is supposed to be constantly moving in the circumference of a circle at the mean rate of 50′′.2 annually, while the revolving pole describes the periphery ABCD in nineteen years, that, if PQ represent the part of the periphery described in that time by the

P

mean pole P about the pole of the ecliptic, the curve line P'Q', a species of epicycloid, will represent the corresponding path of the revolving pole in the same time. This oscillatory motion of the pole about its mean place is designated nutation; and the effects of such movement in changing the apparent right ascensions and declinations of stars are called nutation in right ascension and nutation in declination. That which has been described above depends on the movement of the moon's ascending node; and it is therefore designated the lunar nuta