of the Places of the Stars. 25 



the direction at the time of the maximum, reduced in the ratio of 

 the radius to the sine of R S D, will be represented by the sine of 

 R D ; or of Q K, if R Q be continued till it meet S P, since D Q 

 = R K = S K = 90°, and Q K will be the measure of the maxi- 

 mum. Now, in order to find the point D, we must compute 

 the anglePQD = 180° — PgR^lSO^-PQE — EQK = 

 180° — d—/; and for d we have tang. P Q E = cot. QPE 

 sec. P Q=tang. « sec. w; and in the triangle QEK, EK::=AS, 

 since A E = S K = 90°; but tang. A G = sin. F G tang. A F Gir 

 sin. a tang. <» ,and calling AG:=e, AS = J — e = E K,and tang . 



tang. EK tang. (^— e) tang.(^— e) 



EQK = — : — — — = -— — =: =tang./. 



■^ sm. Q E cos. B E cos. c, 



and FD = 90° — DH = 90°— DQH = d+/— 90° which 

 is the sun's longitude when the aberration in declination va- 

 nishes, and the displacement being directed towards the point 

 90° behind the sun, as he advances from D to C, the declina- 

 tion will be diminished while the sine of © — F D is positive, 

 and its magnitude will be 20",255 sin. (O — d — f — 90°) sin. 



sin. EK sin. ^ — e) 



9, 9 being = Q K, and sin. QK = - — — -— = : — -— 



sm. EQK sm./. 



The values of c and g might also be immediately obtained, 

 from the consideration, that the coefficient of the aberration 

 must initially be the sine of the angle made by the respective 

 great circles with the ecliptic at A and C. 



I. It results therefore from this demonstration, that the aberra- 

 tion in right ascension will be 20,"255 sec. J sin. c sin. (0 — b — 

 90°), and in declination 20 ",255 sin. g, sin (© — d—f— 90°); 

 the values of the coefficients being tang, b rr tang, a sec. a ; cos. 

 c z: COS. a sin w ; tang, d zz. tang, a sec. u ; tang, ezz sin. « tang. 



tang. (^— e), . sin. (^— e) 



m\ tang./, and sm. a = : — - — ; expres- 



cos, c sm / 



sions which cannot easily be rendered much more simple ; al- 

 though it may sometimes be more convenient in calculation to 

 employ the formulas, adopted by Professor Bessel and others ; 

 for the former a' — « — — 20 ",255 sec. i (sin. © sin. a. -|- cos. 

 © cos. a. cos. a) ; and for the latter i' — i — — 20",255, (sin. 

 © cos. a, sin. i + COS. © [sin. « cos. J— sin. « cos. u sin. S\). 



