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$vm = mn \cdot vA / Ar = x \cdot K / T \cdot t / k = K / k \cdot t / T \cdot x = K / k \cdot x$ , nearly. ( $AT$ and $At$ are very nearly in a ratio of equality);

(2)

$∴ vT = vm + mT = K / k x - x = K + k / K x$

Comparing this with the former value of $vT$ , we find

$K + k / K x = a \cdot K 2 - k 2 / K 2$ ;

$∴ x = ka \cdot K - k / K 2 = a / K 2 \cdot k (K - k)$ .

Hence $x$ is at a maximum when $k = 1 / 2 K$ ; then $x = a / 4$ , $mn = x \cdot K / T = AR / AT \cdot qT / 4$ .

Since $mT = 1 / 4 qT$ , $mn = 1 / 4 qz$ ; therefore the diameter of the least circle of aberration is equal to half the lateral aberration of the extreme ray.

Its distance from the focus $q$ is three-fourths of the extreme aberration.

Note. What has been proved here for a lens is equally applicable to the cases of reflexion, and refraction at a single surface, as in both of these, the aberration of a ray inclined to the axis varies as the square of the distance of the point of reflexion or refraction from the axis.

CHAP. X.

REFRACTION AT CURVED SURFACES NOT SPHERICAL.

102. In like manner as in Chap. IV. we found that though a spherical surface is not capable of reflecting light accurately, those belonging to the Conic Sections have that property, so here we shall find that by means of a spheroidal or hyperboloidal surface,