ABE 



ABE 



enters at tli? top, to fall freely through the axis of the tube, 

 wiihout touching the fides of it: and this inclination nuill 

 be greater or lels according to the velocity of the drops in 

 refpeft to that of the tube. In this cafe the angle made by 

 the direttion of the tube, and that of the falling drops, is 

 the aberration arifing from the combination of thcle two mo- 

 tions. 



To find the aberration in latitude and longitude. Let 

 A BCD (AsTRON. pi. i. fi'T. 2.) be the earth's orbit, fup- 

 prfed to be a circle, with the fun in the centre at x, let P 

 be in a line drawn from x perpendicidar to AIjCD, and re- 

 prefent the pole of the ecliptic ; let S be the true place of 

 a ftar, and apcg be the circle of aberration parallel to the 

 ecliptic, and abed the ellipfe into which it is projefted ; 

 let ty^T be an arc of the ecliptic, and PSG a fecondary to it, 

 which will coincide with the Icffer axis bil^ into which the 

 diameter y><jr is projefted ; draw GCxA, which is parallel to 

 fq, and B.vD perpendicular to it muft be parallel to the 

 greater axis ac ; then, when the earth is at A, the ftar is 

 in conjundtiou, and in oppofition when the earth is at C. 

 Eut as the place of the ftar in the circle of aberration is always 

 90° before the earth in its orbit, when the earth is at A, B, 

 C, D, the apparent places of the ftar in the circle will be at 

 a, p, c, q, or in the ellipfe at a, h, c, d: and to find the place 

 of the ftar in the circle, when the earth is at any point E, take 

 the angle yiSj ^ E.vB, and j- Mill be the correiponding place 

 of the ftar in the circle; and to find the projefted place in 

 the ellipfe, draw jii perpendicular to Sr, and i;/ perpendicu- 

 lar to Sc in the plane of the ellipfe, and / will be the appa- 

 rent place of the ftar in the ellipfe; join j- and t, and // will 

 be perpendicular to vt, becauie the piojeftion of the circle 

 into the ellipfe is in fines perpendicular to the ellipfe; draw 

 the fecondary P^l/K, which will, as to fenfe, coincide with 

 I)/, unlefs the ftar be very near to the pole of the ecliptic. 

 Now as <ri)S is parallel to the ecliptic, S and v muft have 

 the fame latitude ; hence vt is the aberration in latitude ; and 

 as G is the true, and K the apparent place of the ftar in the 

 ecliptic, GK is the aberration in longitude. To find thefe, 

 put m and n for the fine and cofine of the angle sSc, or CxE 

 the diftance of the earth from fyzygies, radius being i ; 

 and as the angle svt = the complement of the ftar's lati- 

 tude, the angle vst =r the ftar's latitude, for the fine and 

 cofine of which put i> and Tt', and put r =^ Sa, or Sj : then 

 in the right angled triangle Ssv, l : m : : r : sv =: rm ; and 

 therefore in the triangle I'ts, I : -v : : rm ; tv =^ rvm, the 

 aberration in latitude. Alfo, in the triangle Ssv, 1 : n : : 



rn 

 r : vS = rn; hence w : 1 : : rn ; GK =— ■ ,the aberrationin 



iu 



longitude. When the earth is in fyzygies, m =. 0, there- 

 fore there is no aberration in latitude; and as n is then 

 greateft, there is the greateft aberration in longitude. If the 

 earth be at A, or the ftar in conjunftion, the apparent place 

 of the ftar is at a, and reduced to the ecliptic at H ; there- 

 fore GH is the aberration, which diminilhcs the longitude of 

 the ftar, the order of the figns being <x GT. In this cafe the 



angle A.rE defcrihed by the earth from conjunifiion, or the 

 angle j-Sa, ftiews the elongation of the ftar from the fun. 

 But when the earth is at C, or the ftar in oppulition, the 

 apparent place c reduced to the ecliptic is at Y, and the 

 aberration GE increafes the longitude: hence the longitude 

 is the greateft when the ftar is in oppofition, and Icaft when 

 in coujiindion. When the earth is in quadratures at D or 

 B, then n = 0, and m is greateft, therefore there is no 

 aberration in longitude, but the greateli in latitude ; when 

 the earth is at D, the apparent place of the ftar is at d, and 

 the latitude is there incrcafed ; but when the earth is at B, 

 the apparent place of the itar is at b, and the latitude is di- 

 miniftied ; hence the latitude is leaft in the quadratures be- 

 fore oppofition, and greateft in quadratures after oppofition. 

 Erom the mean of a great number of obfervations. Dr. 

 Bradley determined the value of ;• to be 20". 



From the principles above ftatcd and explained it appears, 

 that the greateft aberration in latitude is equal to 20" multi- 

 plied by the fine of the ftar's latitude; and that the aberra- 

 tion in latitude for any time is equal to 20" multiplied by the 

 fine of the ftar's latitude, and multiplied alfo by the fine of the 

 elongation found for the fame time. The aberration is fubftrac- 

 tive before the oppofition, and additive after it. The greateli 

 aberration in /otiaitudeh equal to 20" divided by the cofine of its 

 latitude; and the aberration for any time is equal to 2o" mul- 

 tiplied by the cofme of the elongation of the ftar, and di- 

 vided by the cofine of its latitude. This aberration is fub- 

 tractive in the firft and laft quadrant of the argument, or of 

 the difference between the longitudes of the fun and ftar, and 

 additive in the fecond and third quadrants. The application 

 of thefe rules will be feen in the following examples: 



1. To find the greateft aberration of y Urfx minoris, whofe 



latitude is 75° 13'. Here m =: I, -u = ,9669 the fin. 75° 



13'; confequently 20" X ,9669 = 19", 34 the greateft 



aberration in latitude. Alfo, « = !,•«;:= >^55' i ""'^ 



20" 



therefore =: 7S", 4 the greateft aberration in longitude. 



)255i 



2. To find the aberration of the fame ftar, when the eartli 

 is 30° from fyzygies. Here m =; ,J, and therefore 19", 

 34 X ,5 = 9", 67 the aberration in latitude. It the earth be 

 30° beyond conjimftion or before oppofition, the latitude is 

 diminilhed; but if it be 30° after oppofition or before con- 

 junftion, the latitude is increafed. Alfo, n = ,866; con- 

 fequently 78", 4 x ,866 = 67", 89 the aberration in longi- 

 tude. If the earth be 30° from conjunftion, the longitude 

 is diminidied; but if it be 30° from oppofition, it is in- 

 creafed. 



3. Eor the fun, m = 0, n = i, and 10 = 1 : hence it has 

 no aberration in latitude, and the aberration in longitude = r 

 = 20" conftantly ; and this quantity of aberration anfwers to 

 the fun's mean motion in 8' 7" 30'", which is therefore the 

 time which the light takes to move from the fun to us at its 

 mean diftance. Hence the fun always appears 20" more 

 backward than its true place. — The following table will ex- 

 pedite the calculations: 



E2 



The 



