1888.] Lord McLaren on an Aplanatic Objective. 371 
consecutive curves. If the intermediate curve be determined 
arbitrarily, this condition can only be fulfilled for definite values of 
/X ; and, indeed, a plane intermediate surface with minimum devia- 
tion brings out an impossible value for one of the refractive indices. 
(3) The crown lens is usually the exterior lens, being the one 
least affected by atmospheric influences. That being so, the first 
surface may be a circle, but cannot be a plane, because a plane 
exterior surface would not secure the necessary convergence. 
(4) If the fourth or inner surface be selected for arbitrary 
determination, it may be a plane, but cannot be a circle consistently 
with the aplanatic conditions ; because, in order that the pole of 
the curve, (r = a cos may fall towards the eye-end, the flint lens 
would have to be a double concave, and the necessary convergence 
would not be attained. 
Comparing these results, it appears that the most simple com 
bination for lenses in contact (but not necessarily the best) is one 
in which the fourth surface is plane, viz., a plano-concave flint, 
placed behind a double convex crown lens ; which agrees with the 
construction of some of the best modern objectives. I proceed to 
find the equations of the curve surfaces for such a combination. 
The analysis is a little complicated, and it may conduce to 
clearness if in the first instance I neglect the refraction at the plane 
surface, and also assume the parameters a-^a^ of equal value. 
I shall afterwards extend the proof to the case of parameters 
determined by the conditions of achromatism, and take account of 
the plane refraction. 
is the refractive index from air to crown : ~ the relative index 
from flint to crown ; = {n-^ -P 1)9 ; Wg = (n^ -1)6 by the nature of 
the curves. 
0 
At the first surface, -P 1)0 : -P 1) — 
0 
At the second surface, <^2 = ^ 2 ^ ' ^2 = “ • 
By the condition of minimum deviation, = <^ 2 ' • 
also c{>j + cfij = Wj -p 0 J 2 . (As before found.) 
oj^ -p (O 2 = 2(f>j = 24 >.j . 
Hence 
