366 Proceedings of Boy al Society of Edinhurgli. [june 4 , 
2n9 = {2n-\- 1 )- ; 
fX 
2n + l 1 
n = 
( 11 ). 
2n ’ ■ 2(/x-l) ' * 
Comparing these values with those found for the single surface 
/(cos nd)j it is seen that where the desired convergence is efiected 
by the double convex lens here found, the value of n is exactly 
half the value of n in the single surface lens of equal parameter. 
Let /X, = 1 -5, in the double convex lens. Then 
n = — 3—— = unity . 
Accordingly, for the value, /x, = 1 *5, the first surface is r = a cos 
the equation of a circle from a point in the circumference as pole. 
The second surface is, r = a sec the equation of a right line from 
the same pole, (12). 
This result (previously unknown to me) is of practical importance. 
It means this; that a convexo-plane splierlcal lens is aplanatic, 
provided the index of refraction of the glass is exactly 1'5. 
Such a lens, when used as an eye-piece (the spherical surface 
towards the observer’s eye), will bring the rays coming from the 
focus into parallelism, so as to be fit for vision. The parallelism 
will be true, within the conditions of my original assumption, viz., 
that the portion of the lens used is so small that arcs may be 
considered proportional to sines. 
In the micrometer eye-piece the condition of simultaneous distinct 
vision of the wires or bars in the focus, and the star, is that the 
two objects shall be seen by pencils of the same aperture, a condition 
which is accomplished by using an eye-stop with a very small per- 
foration. Hence, a single plano-convex lens {p=l '5, and convexity 
to the eye) is an aplanatic micrometer eye-piece. If constructed of 
Brazilian pebble, the index of refraction will be nearly correct, and 
the chromatic and spherical aberration will be alike insensible. 
(2) I proceed to find the values of for the two surfaces of 
an aplanatic double convex lens, under the condition that the ray 
within the lens is inclined at the angle of minimum deviation. 
This condition is contained in the expression 
The parameters a-^a^ as before are equal, whence and by 
the equations of the respective curves, Wj = (?ij -f- 1)^ ; = {n^ -1)0 . 
