164 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
in fact, it was by such a method that I found them ; but their real significance is 
contained in the proof given above. 
We have thus found among the twelve aberrational coefficients six relations which 
may be expressed in terms only of the focal length and other cardinal elements of the 
normal system, or seven, if we include the Petzval expression as of that class. 
Let us consider next what geometrical description can be given of the occurrence 
or absence of the twelve coefficients. It must be remembered that for different 
choice of origins the coefficients do not preserve an identity. Thus if we shift O, the 
original origin to the point ( —c?, o, o), the new set—...—-is given in terms of the 
old set— Siq', ...—by writing in the equations of p. 160. 
S,g = S,h = ... = 0 
and 
g = 1 , h = d, k — 0, I = 1 ] 
and if, on the other hand, we shift the emergent origin to d', we have (^jG, ... 
connected with Sig, ..., which now figures as the old set, as if in the same equations 
we wrote 
(\g' = SJi' = ... = 0 , 
g' = 1 , h' = d', k' = 0 , r = 1 , 
and in the event of both these changes being made a system {g + Sg, ...) is trans¬ 
formed into (G-f-^G, ...), where 
G = g d k, H = fi dg -t- c? ^ -t- dd k, 
K — ky \j — I -\- dk 5 
(^iG = 8 -^g + d'8ik, 
= S-yk, 
(^iL = 8 -Jj + d^Jc., 
(52G = 8 .jg-\-d^-^g^d'8.^y 
J 2 H = 4^' + d {8-Ji 8.£) 4 - dd8^g d'hL, 
4K = ^2^ d^-Jc, 
4L = (4^+4^') 
4G = 49' + 2d49'-t-cP40'+c^^4I^> 
4H = L}i -f d ( 24 /^ ++ d^ (4/i + 24sr) d%g + c^'4L, 
4K = 4^+2d4^'+'^^4^) 
4L = Sji + d ( 24 ^ 4- 4^’) h dk^ (4^ + 24 ^’) + d?^^k. 
