MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 157 
If we compare the scheme 
r 
I + (t), (t)/B 
— (l—+ — eBw, + — CO 
(7), (8) 
with the equations (4), we see the aberrational scheme is the equivalent of a linear 
scheme, for a single surface, for which the plane of origin passes through the point 
x = w/(1—7i)B, that is, through the actual point of incidence cc = |-B(6o^ + Co^)••• 5 
if we take the curvature as B (l +|-eg^ + |-e7’^), and the refractive indices u (1 — 
/(l — |-y'^) respectively. 
As to the latter it may be noticed that the exact equation for refraction of a ray 
impinging at the origin, and in the plane Ox^, is 
or, to our order. 
ju sin (tan ^ /B) = n' sin (tan ^ /3') 
M(l-i/3^)./3 = M'(l-ir)-iS- 
Hence the aberration may be described as due purely to the obliquity of the ray 
to the axis, and the aberration w to the lateral separation from the axis, and we see 
that the somewhat remarkable fact that two functions co, suffice to express the 
aberrations of every ray may be stated in the form that there is no term which is 
produced jointly by obliquity and lateral separation. 
If in any instrument we have a number of surfaces each introducing aberrational 
terms, and if the schemes preceding and following the surface (r) be compounded so 
as to read, say, {g, ...}, {g', ...}, then the whole may be represented by 
1 + 
0) 
/B. 
— (l — 71,.) B,.4-B,.'»kr“frBrW;., 71,. +—J 
( 10 ) 
and the portions added to the general scheme in consequence of the aberrations of 
the surface will be 
r 
SlJ 
[g.h 
-ij U'', U, l] LB. ij U-', I'} 
and in this the schemes may be taken at their “normal” values 
without regard to aberrations introduced by surfaces other than the surface (7’). If 
then we write, adding the effects of all the surfaces, 
.( 12 ) 
and if we now denote by {^, ...} the scheme got by compounding all the normal 
schemes of the instruments in succession, whether these are refractions, or mere shifts 
