Investigating Aberrations of Symmetrical Optical System. 503 



Hence the identit} r u — ly — mx becomes u~y ; so v = 2. 

 Substituting for ?/, z in the equations above, we have 



m' = m//jb—(l — l/fjL)u/r; n'=njij,— (1 — ljfjb)v/r ; 



h' = u//jl+(1 — l/fi)u ; v' = i'l fi+ (1 — 1/ /jl)v ; 



i. e. u' — u and v' = v. 



It is clear from this result and from general considerations 

 of symmetry that it is sufficient to restrict attention to two 

 only of these equations, say those for m' and u' . The other 

 two are of precisely similar form and can be at once written 

 down, replacing m, m!, u, v! by n, n' , v, v' respectively. 

 From this on we shall then consider only the transformation 

 effected by refraction in the two coordinates m and u. The 

 transformation in n and v is then known, mutatis mutandis. 



The first approximation is thus represented by 



m'=m/fi-(l-l/p)u/r (12) 



u'=u (13) 



(5) The Second Approximation. 



For the second approximation we can no longer write 

 cos = 1 ; instead we must set cos # = 1 — -J sin 2 6. 



cos^l-lsin^'^l-^sin 2 ^ 



Equation (10) gives 



sin 2 6 = -I a 2 + (u 2 -f v 2 ) + 2r(mu + n v) + r 2 [m 2 + n 2 ) \ ; 



here a 2 is of the fourth order and is to be rejected. 

 Thus 



cos 0'-(cos 0)//x= l-l/fjL+ i(l/{i-l//j, 2 ) sin 2 6 



= G*-l)fr + i(/.- l)/v 2 {(u* + v 2 )/r 2 



+ 2 (mu + nv)\r -f {m 2 + n 2 ) } . 



Again the equation to the sphere is 



(a?-f) 2 4j/ 2 + ^ 2 =r 2 , 

 i. e. 2rx = x 2 + y 2 + z 2 . 



Hence to lowest order 



x = (y 2 + z 2 )l2r. 



But also to lowest order y = u, z = v. 

 Thus to the second order 



. x={u 2 + v 2 )j2r. 



