Investigating Aberrations of Symmetrical Optical System. 501 



But since we have defined line-coordinates a = ny — mz, 

 etc., we can write these equations as 



a' = a//j>; 1 



V = b/ p + z{cos 0'- (cos 0)/ /*}•,[ ... (7) 



c' — c/fi— y {cos 6'— (cos 0)//jl}. ) 



Equations (6), (7) are equations which give the coordinates 

 o£ the refracted ray in terms of the coordinates of the incident 

 ray and also of the coordinates of the point of incidence and 

 of the angles 6, 6' of incidence and refraction. 



Oar next task is to express a?, y, ~, 6, 6' in terms of I, m, n, 

 a, b, c. 



Before so doing we may conveniently call attention to the 

 first equation of (7), namely a =a//jb. Writing it as a'fjL — a, 

 we see that the moment about the optic axis of the ray in 

 any medium, multiplied by the refractive index of the medium, 

 is invariant for a single refraction and hence for any series 

 of refractions at coaxial spherical surfaces. In other words, 

 the moment of a ray about the optic axis of an optical 

 instrument varies inversely as the refractive index of the 

 medium in which the moment is measured. It should be 

 observed that this is true exactly and not merely as an 

 approximation. This fact leads to certain useful identities 

 subsequently stated. 



We may also observe that the two fundamental identities 



I 2 -f- m 2 + n 2 = 1 and at + bm + en = 



are sufficient to fix I and a, when we know the four 

 quantities m, n, b, c. It is sufficient therefore to work in 

 terms of m, n, b, c only : from a knowledge of these the 

 position of the line is completely determined. It is to be 

 remarked that it is only the possession by the optical instru- 

 ment of a rotational symmetry about its axis that enables us 

 to discard ^ and a and work in terms of the remaining four 

 coordinates alone without serious violation of symmetry. 



Reference to the second and third equations of (7) shows 

 that this symmetry is better served, if we work with — c, 

 and b rather than with b, c as corresponding to m, n. We 

 therefore write 



u= — c ; v = b. 



Our defining equations are now 



a = ny — mz : u = ly — mx : v=-lz — nx : 

 and the identity al — un — vm (8) 



