the Five Aberrations of von Seidel. G81 
case of special interest is that in which x, y, z and x\ y\ z' 
are conjugate points, i. e. images of one another in the optical 
system. The ratio x : x' must then be independent of the 
special values ascribed to I, V . In order that this may be 
possible, L e. in order that z, z' may be conjugate planes, the 
condition is 
(^ + 2P)(y-2P0 + Pi 2 =0, .... (8) 
and then 
x y z + 2? P 1 
(9) 
x'~if~ P x ~z'-2F n ' ' 
giving the magnification. 
Equations (8), (9) express the theory of a symmetrical 
instrument to a first approximation. In order to proceed 
further we should have not only to include the terms in (7) 
arising from T (4) , but also to introduce a closer approximation 
for n. Thus even though T (4) = 0, we should have additional 
terms in the expressions for x, x' equal respectively to 
ilz(l 2 + m 2 ) and ^V(7 /2 + m' 2 ). 
If the object is merely to express the aberrations for a single 
pair of conjugate planes, we may attain it more simply by a 
modification of Hamilton's process. 
Supposing that the conjugate planes are s = 0, z' = 0, we 
have V a function of the coordinates of the initial point %', y', 
and of the final point x, y. And if as before I, m, n, V, m!, n 1 
are the direction-cosines of the terminal portions of the ray, 
we still have 
l = dY/dx, m = dY/dy, . . . (10) 
V=-dV/dx\ m'=-dV/dy'. . . (11) 
But now instead of transforming to a function of /, m, 
V, m\ from which a?', y\ x, y are eliminated, we retain x\ y' 
as independent variables, eliminating only x, y, the coordinates 
of the final or image point *. For this purpose we assume 
■U = ls + my-V (12) 
The total variation of U is given by 
dJ] = xdl-rldx+y dm + m dy 
dV , dY . dN . . dV, t 
^— ax ^— ay —. ax =— f ay' 
dx dy J dx' dy' ^ ' 
or with regard to (10), (11) 
d~U == x dl +y dm + V dx' + m' dy' ', . . . (13) 
* Compare Eouth's ' Elementary Rigid Dynamics,' § 418. 
