MR. E. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 155 
In the same way it follows 
c' — (’[l+w] +y[«/B], 
y = C 
C 
— (l — n)B + Bx/r — ^ w 
■f y \_n + — w] ; 
( 8 ) 
for the case of the paraboloid 
sphere 
We shall generally write 
C/B^ = 0, 
C/B^ = B. 
C/B^ = eB. 
We remark that the coefficients that transform the {h, system into (I/, ^') are 
the same as those which transform (c, y) into (c', y'), and for any surface each is 
expressed in terms of the two functions xj/-, « defined by equation (6), in addition to 
the refractive index and curvature. 
These equations therefore permit us to treat rays which cross the axis with the 
same readiness as those which intersect it, a thing which is very troublesome in the 
trigonometrical discussion of the question. They also apply equally easily to the 
sphere, the paraboloid, and any intermediate form. 
Before proceeding with the discussion of these formulae I shall verify that they 
cover the known expression for longitudinal aberration on the axis after refraction at 
a single spherical surface, as it is given in the text hooks. 
Suppose the ray meets the axis at a: = x' = v', so that 
h + ^v = 0, + = 0 ; 
then the equation connecting v, v' for the case of the sphere is 
— lu/[— (l —7^.) B + B (\/r —ft))] + y'[w + l/r —w] —[l+w]+w/B = 0, 
or, dividing by vv' and rearranging the terms. 
but 
and 
also 
^ = in(r-/3^) = - 
\v vy \v vj \v V 
B-~ B-i 
_ V _ _ _ 'ly _ V . 
n 1 1—n 
X 2 
