152 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
H; Hz _F 2 § 
N (5 F, H, h1 
Fig. 3. 
Set the axes as shown in the figure, the distance lietween the points H]H'i and 
being annulled. Then the lines Fslij, F'jib S'^ve T>, T>', the principal foci, and 
the line iiiiia gives points N, N' which are conjugate to one another and are the nodal 
points of the compound system. 
We see that it is always possible to determine a geometrical system that shall 
correspond to any given values of p, A, k, 1. Thus, for example, n — 0 Implies that 
F' is conjugate to every point of the original system, or, what is the same thing, that 
every emergent ray goes through Ft 
If the emergent origin is at the principal focus, (7 = 0 . 
If the original origin is at the principal focus, / = 0. 
If the original and emergent origins are conjugate points, h — 0. 
We shall now consider the case of refraction of a general ray at a symmetrical 
surface centred upon the x-axis and sliall show that a scheme jp + dp, ...{ maybe 
derived for it, whicli shall Include the aberrations; these, represented by the 
additional terms ^g, will, of course, vary from point to point with the squares and 
products of the co-ordinates and angles of incidence upon tlie surface, whereas for the 
pure linear scheme g, ... are the same for every ray of the beam. 
Taking rectangular axes Oxyz, let the ecpiation of the siuTace separating the region 
of index v from that of index v be 
'2x = B + .( 5 ) 
Let a ray 
y = ^x + h, z = yx + c, 
in the original medium be transformed by refraction at the surface into 
y = /3V + ?/, 3' = yV + 0', 
where the axes are, in fact, the same but are accented to indicate the difference of 
medium. 
The positive direction of the x-axis is that in which the light is travelling. 
