2 F. J. CHESHIRE ON ABBE'S TEST OF APLANATISM, 



fx and /xi, respectively, and if m equal the magnification produced 

 by the system, the sine-law fully stated takes the form — 



sina = M tH = k (a constant) * .... (1) 

 sin /3 /jl 



In general, a ray, passing from the point p to the point Q, and 

 undergoing a total deviation equal to the sum of the angles a 

 and /?, would suffer, in any practical optical system, many 

 refractions, which, however, it is not necessary for our purpose 

 to consider. All that we are concerned with is the total devia- 

 tion, and this may be looked upon as though produced by a 

 single refraction only, at the point c, obtained by producing 

 the incident ray and its conjugate ray until they meet as shown. 

 To the point c the name chief point has been given by Professor 

 S. P. Thompson. 



In a similar way, let Ci be the point of intersection of a second 

 pair of rays, making angles a\ and/?!, respectively, with the axis. 

 Then from simple geometry we have — 



sin a c q 

 sin (3 c p' 

 and — 



sin ai Ci Q 



sin fii Ci P ' 

 and, since the ratio of the sines is constant, — 



c Q _ Ci Q, / 2 \ 



C P Ci P 



and so for any pair of rays. It follows from the constancy of 

 this ratio and the fixed distance p Q, that the chief points must 

 lie upon a curve, which is the locus of a triangle constructed on 

 a given base and with a constant ratio between the lengths of 

 its other two sides. This locus is a circle, f with its centre on 

 the axis P Q, and cutting it, say, at v. Let p v = a, and v Q = b ; 

 then r, the radius of this circle, is obtained from — 



r = ^; (3) 



o — a 

 and, if a be less than b, the centre o of this circle is at a point 

 on the axis such that we have for d the distance p o : 



d^^~ W 



o — a 



We have thus arrived at the following important result : — In 



* See HeatWs Geometrical Optics, 1887, p. 255. 



\ Briggs and Bryan, Co-ordinate Geometry, 3rd edition, Part I., p. 186. 



