the Five Aberrations of von Seidel. 683 
In order to complete the value o£ x we must add the 
expressions in (19) and (21). 
Since F disappears from the values o£ x and y, we see that 
there are jive effective constants o£ aberration of this order, 
as specified by SeideL The evanescence of A is the Eulerian 
condition for the absence of spherical aberration in the 
narrower sense, i. e. as affecting the definition of points lying 
upon the axis (x' — Q). If the Eulerian condition be satisfied, 
B = is identical with what Seidel calls the Fraunhofer 
condition *. The theoretical investigation of this kind of 
aberration was one of Seidel's most important contributions 
to the subject, inasmuch as neither Airy nor Coddington 
appears to have contemplated it. The conditions A = 0, B = 
are those w T hich it is most important to satisfy in the case of 
the astronomical telescope. 
To this order of approximation B = is identical with the 
more general sine condition of Abbe, which prescribes that, 
in order to the good definition of points just off the axis, a 
certain relation must be satisfied between the terminal 
inclinations of the rays forming the image of a point situated 
on the axis. The connexion follow r s very simply from the 
equations already found. Bv (15), (16)," (17)/(20), with 
m = 0, 
I' = MZ + B/ 3 + terms vanishing with %', y ] ; 
so that for the conjugate points situated upon the axis 
Z' = M/ + B/ 3 (23) 
The condition B = is thus equivalent to a constant value 
of the ratio V/l, that is the ratio of the sines of the terminal 
inclinations of a ray with the axis. And this is altogether 
independent of the value of A. 
On the supposition that the two first conditions A = (3, 
B = are satisfied, we have next to consider the significance 
of the terms multiplied by C and D. Since 
dxjdl = (V 2 , dyjdm = Dx"\ 
we see that C and D represent departures of the primary and 
secondary foci from the proper plane. In fact if 1/pi, l/p 2 
* If A be not equal to zero, it can be shown that the best focussing of 
points just off the axis requires that 
A/ +Rr'=0, 
where l is the value of I for the principal ray. For example, if the 
optical system reduces to a combination of thin lenses close together, 
l =zx/f, where f is the distance of the lenses from the image plans. 
Since by (19), x=Mx', the condition may be written 
AM+B/=0. 
