MR. E. A. SAMPSON; A NEW TREATMENT OF OPTICAL ABERRATIONS. 159 
In these expressions the values of (^iG, ... are not unrestricted. Thus, for example, 
the rays which originate in the point {h, c) must upon emergence be normal to a 
surface. Consider the conditions that 
y' = (3'x' + h', 7 J = yV + c', 
where V, c', y are functions of two variables /5, y, as above, should be normal to a 
surface. 
If {x\ y\ 2 ') is a point upon the surface, then we have for all directions upon the 
surface 
dx' -V (i'dy' y dz' = 0 , 
or, since x\ y\ z' are functions of /3, y only, 
Also 
Therefore 
ox' r^t cy' r 92^ _ A 9''^' , or clf , oz! 
y’ = fi'x' + h', z' = yx' + c'. 
ex' 
9/3 
ox! 
,, I 0^ , . f f f ^y' ■ 9c'' 
9y 9y 
(1 +/3'^ + r'n +/?'(*') + r'(a:'gi + ^) = 0 ; 
oy Oy/ 
or, say. 
A {®'(l+/3«+rT‘'} + (/3'P^ H-y'^)(l+/3'"+yT* = 0, 
Oy \ oy Oy / 
SO that the necessary and sufficient condition is 
vn ? /a' 
3 O. y) 3 (/ 3 , y) 
Retaining only the terms of lowest order we liave 
^h' 
b' 
^ = H T— = (^ 2 ^ • ^c + ^G . by + . c/3 + (^gH . /3y, 
op Oy 
98' j 
9/3 
9c' 
. 9c+ (^ 3 !^ . 9y + <l.X . + . /3y, 
oy 
= (^ 2 G . be + f^gG . cj3 + ^ 2 ld • by + (^gll . /3y 
op 
= ^ 2 ^- • 9c + ^gK . c/3 + 3' ^sR • /^y 
op 
oC 
Oy 
= H, 
= R, 
oy 
