The above value of R is quite exact enough for all observations as far as 80° from the zenith. From about 80° to the horizon the changes of refractions are so uncertain that observations are useless for the nicer purposes of astronomy. The following tables will be found very convenient for computing the quantity of refraction for all the zenith distances not greater than 80°. For the particulars of the construction of these tables, and for several investigations relative to astronomical refractions, references may be had to the 12th volume of the Transactions of the Royal Irish Academy; in which I have deduced the above expressions for refraction, independent of any hypothesis relative to the variations of density in the atmosphere. By help of Table I. the first term of R is obtained. The second table gives the second term of R, which near 80° has been slightly modified. TABLES FOR REFRACTION. TABLE I. Far. Loga- Far. Loga Far. Loga Therm. rithms. Therm. rithms. Therm. rithms. Logarithm in Table I. + log barom. + log tan zenith dis tance log approximate refraction. Appr. ref-Number Table II. = refraction. Example. Zenith dis 71° 26', barom. 29,76 inches, and ther. 43o. Prop. XVIII. From given small variations in the longitude of a celestial object, and in the obliquity of the ecliptic, to deduce the variations of the right ascension and north polar distance. 1. For the effects of the variation in longitude. Let P (fig. 62. 1) be the pole of the ecliptic, N that of the equator, and F the fixed star or other object. Let FPM the increase of longitude, the distance PM from the pole of the ecliptic remaining the same. Then FNM is the increase in right ascension, and drawing MR parallel to the equator, FR is the decrease of north polar distance. FNM RM:: rad: sin MN RM FM:: sin RFM cos PMN : rad sin PNM.cot PMN cot PN. sin NM- cos PNM. cos NM. Hence by substitution FNM-FPM. (cos PN-sin PN. cos PNM. cot NM). (a) -FPM. sin PM. sin PMN FPM. sin PN. sin PNM. (b) 2. For the effects of the variation in the obliquity of the ecliptic. Let AP, PS (fig. 62, 2) represent the right ascension and declination of the point S, and AQ, QS the right ascension and declination when the obliquity of the ecliptic is changed by the angle PAQ. Then when this angle is very small, PR= the change of north polar distance nearly, and RQ= the change of right ascension nearly. a Trig. p. 69, art. 112. sin RP sin A:: sin AP: sin ARP, or RPA. sin r. ascension nearly. (c) Also cot AR tan RP = cos ARP=tan RQ. cot SR, hence RQ= RP. cot AR. tan SR, and RQ A. cos (rt. asc.). cot (north polar distance). Application of the preceding Prop. (d) The above proposition enables us to deduce the apparent right ascension and north polar distance from the mean, as affected by precession, lunar nutation and solar nutation. The actions of the sun and moon causing a change of place of the intersections of the ecliptic and equator, the longitude of each object is changed by the same quantity. The same action also occasions the obliquity of the ecliptic to be variable. The action of the planets also changes the plane of the earth's orbit, and therefore the intersections and inclination of the ecliptic and equator. Let L the mean longitude of a star in the beginning of 1820. Then for the time 1820+t, t being taken negatively or positively, App. long-L+50", 19.t-17",30. sin long ▷'s node +0",21. cos 2 long >'s node-1",25. sin 2 long-0",21. sin 2 long D. (e) App. obliq. of ecl. (O), for 1820 + t = 23° 27′ 47′′ —0′′,45. t + 9.25. cos long D's node-0",09. sin 2 long D's node +0",54.cos 2 long +0",09. cos 2 long D. (f) The form of these quantities has been obtained by investigations in physical astronomy, the larger coefficients and 0",45 have been obtained by observation. In the above the term 50", 19t increased by 0", 13t serves for determining what is usually called the effect of precession in right ascension and north polar distance. When so increased it is the mean effect (luni-solar) of the sun and moon. The part 0", a Art. 90, &c. T |