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therefore, the angle instead of its sine (App. 51) and assuming cos r = 1, we have,

tang R +2


=m tang R,


r = (m-1) tang R.


Now in the triangle CAa we have, Ca: CA :: sin CAa or sin ZAS': sin CaA, or ę + h:ę :: sin Z: sin R. Hence, sin R sin Z ę = s+h



Taking m = 1.000284 (78) and substituting for its value 206264".8 (App. 51), we have, (m—1). w = 58′′.6. Hence, since = 3956 (71) and h ę = 5.13 (77,) the formulæ (C) and (D) become,

sin R

3956 3961.13


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sin Z


r = 58".6 tang R

The degree of accuracy of these formulæ may be tested by finding the latitude of a place from the observed upper and lower meridian altitudes of different circumpolar stars (58), using the formulæ in computing the refractions; which must be subtracted from the observed altitudes to obtain the correct altitudes. If the state of the air is the same or nearly the same as that assumed in finding the formulæ, and if no one of the lower altitudes of the stars employed is less than about 20°, the latitude as obtained from different stars will be sensibly the same. But if the lower altitude of any one of the stars is much under 20°, the latitude found from that star will be decidedly too great. Whence it follows that for a low altitude, the refraction computed by the formulæ is too small. It may thus be ascertained that for altitudes of 20° and upwards, the refractions computed by the formulæ do not err to the amount of a second; but for lower altitudes the error becomes considerable, amounting at the horizon to several minutes.

83. Tables of Refraction. The complete investigation of astronomical refraction is a subject of great difficulty. It

has claimed the attention of many eminent mathematicians,* and formulæ have been obtained which give the amount of the refraction with great precision, except for altitudes under 12 or 14°; and for these they give it very nearly. These formulæ take into view the changes in the density of the air at the earth's surface as indicated by the barometer and thermometer. From the formulæ, tables have been computed, from which the refraction corresponding to a given observed altitude is easily obtained. In these tables, the principal columns contain the refractions computed for a density of the air corresponding to some medium heights of the barometer and thermometer. These are called mean refractions. Other columns contain the corrections due to given changes in the states of these instruments.

84. Refraction increases the visible continuance of the heavenly bodies above the horizon.

As refraction increases the altitudes of the heavenly bodies, it must accelerate their rising and retard their setting, and thus render them longer visible. The refraction at the horizon is about 34', which is rather greater than the apparent diameter of the sun or moon. Either of these bodies may therefore be wholly visible when it is really below the horizon.

85. Oval form of the discs of the sun and moon when near the horizon. This is an effect of refraction. As R must be nearly equal to Z, (82. D), and as the tangent of an angle increases rapidly when the angle approaches to 90o, it is evident from the expression for r (82. D), that the refraction must increase rapidly near the horizon. Hence the lower part of the disc, when in that situation is considerably more elevated by refraction than the upper; and consequently the vertical diameter and chords parallel to it are shortened, while the horizontal diameter and its parallel chords are not sensibly affected. This necessarily

* Laplace, in the Mécanique Céléste; Prof. Bessel, in the Fundamenta Astronomiæ ; Dr. Young, in the Transactions of the Royal Society of London for 1819 and 1824; Ivory, in the same Transactions for 1823; and various others.

causes the disc to assume an oval form. The apparent diminution of the vertical diameter amounts, at the horizon, to about of the whole diameter.

86. Apparent enlargement of the discs of the sun and moon when near the horizon. Although this is not an effect of refraction, it may properly be noticed here. It is an optical illusion of the same kind as that which makes a ball or other object appear larger when seen at a distance on the ground than when viewed at the same distance from the eye, on the top of a high steeple. Our judgment of the magnitude of a distant object depends not only on the angle it subtends at the eye, but also on a concurring though sometimes very erroneous impression with regard to the distance; the same object, seen under the same angle, appearing larger as there is an impression of greater distance. Now in viewing the sun or moon when at or near the horizon, the various intervening objects near the line of sight, give the impression of its being more remote, than when seen in an elevated position. When the sun or moon is viewed through a smoked glass which renders intervening objects invisible, the disc does not appear thus enlarged.

87. Twinkling of the Stars. From changes in the temperature, currents of air and other causes, the atmosphere is continually more or less agitated. This agitation produces momentary condensations and dilatations in its constituent molecules, and thus occasions slight but sudden and continually repeated deviations in the directions of the rays of light which traverse it. As the stars appear merely as luminous points, presenting scarcely any visible discs, these irregularities in the directions of their rays of light give to them the apparent tremulous motion called the twinkling of the stars.

The discs of the planets, though small, are much larger than those of the stars, as is shown by observations with the telescope. They are therefore less affected than the

stars, and the twinkling is but little observable in them except sometimes near the horizon where the cause producing it usually acts with the greatest effect.

88. Twilight or Crepusculum. This depends on both reflections and refractions of the sun's rays in the atmosphere. When in the evening the sun has descended so far below the horizon as to cease to be visible by refraction (84), a portion of the lower part of the atmosphere ceases to receive his rays directly, and is only illumined by light diffused through it by reflection from the higher parts. As the sun continues to descend below the horizon the part of the atmosphere that is not directly enlightened by his rays increases, and at the same time its illumination gradually diminishes in consequence of the diminished portion of the atmosphere from which its light is received. This gradual diminution of the light continues till the sun has descended so far below the horizon as to cease to illuminate any sensible portion of the atmosphere above it. This takes place when he is about 18° below the horizon. The last appearance of twilight must evidently be in the western part of the heavens.

In the morning the twilight commences, or the first dawn of day is perceived in the eastern part of the heavens when the sun has arrived within about 18° of the eastern horizon; and the light then increases in the same gradual manner as it diminishes in the evening.



89. Apparent and true places of a body. The place which a planet or any other of the heavenly bodies, except the fixed stars, appears to occupy in the celestial sphere


varies with a change in the position of the observer. tronomers, therefore, in order to render their observations easily comparable, and for convenience in computations, reduce the place of a body as observed at any place on the surface of the earth, to that in which it would appear to be, if seen from the centre.

The place in the celestial sphere in which a body would appear to be as seen from any point on the earth's surface if there were no refraction, is called the apparent place of the body; and that in which it would appear to be if seen from the centre of the earth, is called the true place. Thus if C. Fig. 15, be the centre of the earth, A a place on its surface, Z the zenith of this place, B the place of a body, and b and c the points in which AB and CB produced, meet the celestial sphere, then is b the apparent place and c the true place of the body.

90. The Parallax or Parallax in altitude of a body is the angle contained between two straight lines conceived to be drawn from the centre of the body, one to the centre of the earth and the other to a place on its surface. Thus, for the place A, the angle ABC is the parallax of a body at B.

The Horizontal Parallax is the parallax when the body is in the horizon, or, which is the same, when the apparent zenith distance is 90°. Thus, the angle AB'C is the horizontal parallax of the body.

The Equatorial Parallax of a body is its horizontal parallax for a place at the equator.

91. The parallax of a body is equal to the difference between the apparent and true zenith distances of the body, or between the true and apparent altitudes.

For as ZAB is an exterior angle of the triangle ABC, we have ang. ZCB+ ang. ABC= ang. ZAB; or ABC = ZAB ZCB. But ABC is the parallax, ZAB the apparent zenith distance, and ZCB the true zenith distance. As the altitudes are the complements of the zenith distances, the difference between them must be the same.

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