156 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
Hence the right-hand member above is equal to 
This is one of the usual expressions; compare Herman’s ‘Optics,’ p. 189, (iii.). 
After a slight transformation it leads to the Zinken-Sommer expression for the 
separation of the focal lines in any co-axial system, and thence, as Whittaker has 
shown (‘ Theory of Optical Instruments,’ p. 26), to the expressions of Seidel’s 
theory. 
We may verify also that these expressions lead to the known results in the case of 
the parabolic mirror. 
Consider the focus for rays parallel to the axis, lc,, when j3 = y = 0. 
0 = 1-fw + i/[—(l —n) B-t-Bi/r], 
But 
71 = —I, w = + + 
yf, = ^7i{/3'^+y'^) = ^i{l-nyB^b^+c^) = -2B^ {h^+c% 
so that 
v' = [l+B^{b^ + c^~)]/[2B + 2B^¥ + c^)] = 1/2B, 
so that the longitudinal aberration vanishes at the principal focus. More generally, 
the ray y' = I3'x + b', ^ = y'x-^c\ corresponding to a general incident ray for which, 
say, y = 0, meets the plane x’ = d' in points whose co-ordinates are 
y = 6 [— (l —n) Bc2'-l-1 -\-Bd'+^\_d' {n + xj^—u)) -f-w/B], 
2 '= c [ — (1 — n) Bd'-t-1-t-Bd'i/r+w]. 
For the paraboloidal reflector 
o) = B^b^ + c% 
xjr = + ^ -2B^{W+c^)-2Bbl3, 
since /8' = —/3—2B6, y = 2Bc; therefore taking the focal plane d' = l/2B, we have 
y = -^l2B-Bb‘^^-^i^B{b‘^+c^)+b^l 
z' = — B6c/3 ; 
(9) 
these are known expressions, leading to the theory of the coma of a parabolic 
reflector; cf. Plummer, ‘ Mon. Not. B. A. S.,’ LXIL, p. 365, (9), (lO). 
