686 Hamilton' s Principle and the Five Aberrations of v. SeideL 
Hence, by (14), 
x = 2*1 + yx\ y = 2/3m + By'. 
If x, y is conjugate to x' , y\ we must have 
a = 0, 13 = 0; 
so that 
* = <r*, y = V- . - . . (30) 
These are the equations o£ the first approximation, and 
they indicate that the magnification need not be the same in 
the two directions. 
There are no terms in U {3) . As regards U (4) , we have 
U< 4 > = AZ 4 + BZ 2 m 2 + Cm 4 
+ Vx'P + Ex'lm 2 + FyW + GyTni 
-f B.x ,2 P + la/ 2 ™ 2 + J^y7m + Kz/ 2 / 2 + Ly'*™" 
+ Mx'H + NVy m + Oafy' *J + Py ,s m 
+ terms independent of I and ??z (31) 
In (31) there are 16 effective constants as compared 
with 5 in the case where the symmetry round the axis 
is complete ; so that such symmetry implies 11 relations 
among the constants of (31). For example, in the terms 
of the first line representing Eulerian aberration, axial 
symmetry requires that 
C = IB = A (32) 
We will next suppose that the only symmetry to be 
imposed is that with respect to the primary plane y = ; 
so that U is unchanged if the signs of y' and m are both 
reversed. U (2) is of the same form as in the case of double 
symmetry, and 
x = 2al + yx' s y = 2/3 m -f By'. 
If x, y is the image of %', i/, formed by rays in both 
planes, a = 0, j3 = 0, as before. But it may happen, e. g. in 
the spectroscope, that there is astigmatism even in the first 
approximation. If the points are images of one another as 
constituted by rays in the primary plane, a = 0, but /3 is left 
arbitrary. 
The next term in U may be denoted by U (3) . If no 
conditions of symmetry were imposed, U (3) would include 
16 effective terms, i. e. terms contributing to x, y ; but the 
