MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 165 
But let us defer discussion of these and examine two particular cases of special 
Importance, namely, let us assign meanings to (^jG, ... : (l) where the emergent origin 
is the principal focus, so that G = 0, and therefore ^G = (^jH, and (2) where the 
original and emergent origins are conjugate, so that H = 0, and therefore (^gG = 4H- 
In the former the original origin may be anywhere, but may conveniently be supposed 
to lie at the tangent plane to the first refracting surface. The original rays are in 
constant direction, so that we mav take 
^ «/ 
= const., y = 0, and, say, h = d cos <p, c = d sin <p. 
Then if we receive the emergent ray on the plane parallel to O't/z' which passes 
through a point slightly removed from O', say at 
and it cuts this plane at y' = h' + Sb', ^ = c' + (^c', we have 
V-vW = + [H + U/']/3 
+ ^d cos 0 [c?^^iG + 2d/3 cos + 2dl3 cos + /3^4H], 
c' + ^c' = [* + K4/’']c 
+ \d sin <p [dI^^iG+2c?/3 cos ^^gGr + zd^^sG]!..(23) 
Let us take 
U = (H + L(^/')/3, c' = 0, 
so that 
SV = 1/3 [d^ (,5iH + 4G) + (3%IL] 
+ cos </)(i [K4Z' + |-(i^^iG+|-jd^ ((53G + 24H)] 
+ cos 2({)(P [^yddgG], 
<5c' = sin <pd{KSf'+^d%G + ^^%G'] 
+ sin 2<l>d^ [i-/d(52Gr].(24) 
These express the amounts by which the aberrations disturb the ray from its 
normal focus. Consider the lines in turn and examine their significance when the 
original ray traces out a circle d = const. 
The terms 
i/3[(^M<^iH + 4G)+/3^^3H] or ^^[2d%G+^%H.'] 
give a fixed point. It may be considered as adding to the focal length 
the terms 
R + Uf' 
