STARS IN COMA BERENICES. 439 



the heavens. A certain quantity, known as the "Transforma- 

 tion Correction," must therefore be added to reduce any meas- 

 ured distance on the plate into the distance on the sky to which 

 it corresponds. To find an expression for this correction, let 

 us consider the spherical triangle whose vertices are the pole, 

 the center of the plate, and any star on the plate. By center is 

 meant the point at which a perpendicular on to its plane from 

 the object glass cuts the plate. It is the point of tangency of 

 the plate with the spherical image of the sky formed at the 

 focus of the object glass. Now let 



a^ = the right ascension of the center, and 



a = the right ascension of any star ; 



/q = the north polar distance of the center, and 



/ = that of the star ; 



7j = the parallactic angle at the center, and 



X = the angular distance from center to star ; 



then, by the usual formulae [Chauvenet, Sph. Trig., Equ.'s 

 (122), (123)] 



cos (/(, (/) COS/ 



COS (^ 



(I) 



sin (;>„ — i?) cot (a — aA 



cot 77 = ^^ iJ !^ °l ( 2 ) 



sin (^ ^ ' 



where 



tan ^ ==: tan / cos ( a — a^). (3) 



Now consider a central projection of the figure onto a tangent 

 plane at the center of the plate, O. Let OX, OV he the axes, 

 OY being the projection of the hour circle through the center 

 0, and OX being perpendicular to OY. Let also 5 be the pro- 

 jected position of the star, and A^ and Y its rectangular co- 

 ordinates on the plate expressed in seconds of arc of a great 

 circle, the positive directions being the same as those of the 

 "Measured Coordinates" (cf. p. 427). Then we shall have, 

 taking the radius of the sphere as unity, 



X= OS s\n YOS 



¥--=. OS cos YOS. 

 (99) 



