1888 .] Lord McLaren on an Aplanatic Objective. 
369 
As the rays after refraction at the two surfaces of the crown lens 
converge to the pole, a-^ is the focal length of the first lens, which 
is given by the achromatic equation of condition; — a-^ = a 2 =fi. 
is measured from the same pole ; hence less the distance 
between centres of lenses ; or ^3 - A . 
The rays after refraction at the fourth surface converge to the 
principal focus of the telescope; hence «4 = F less the distance 
between the centres of the lenses ; or = F — A . Thus the four 
parameters are immediately found. 
To express 6^ in terms of ^3 ( = 6j) we have (under the original 
convention whereby arcs are considered as proportional to sines), 
arc^ — arc^, or a^O^ — a^O^, whence 0^ = 0q. ^ . 
a^ 
The ratio ^ may he denoted by X . 
«4 
The equations of the surfaces of the first lens have already been 
found (Eq. 13). They are those of a double convex lens with 
parameters ^^ = ^ 2 , the surfaces being respectively of the forms 
/(cos n^Oj), /(sec n^Oj), and having the ray within the lens inclined 
at the angle of minimum deviation. 
As there found, — 
2 - u, 
n-, = : n. 
2/XJ-2 
The equations of the second or flint lens are to be found in terms 
of Oq and 0^. As the lens is concavo-convex, the foci are exterior 
to the curves, which accordingly are of the form /(sec nO). 
Observing that the rays incident on the concave surface of the 
flint lens converge to the pole common to it and the crown lens, 
and treating the rays proceeding from the convex side of the flint 
lens as a diverging incident pencil, we have 
<#•3 = » 3^0 : <I>S = : ^4 = »A = ■ 
/^2 /^2 
As the lens is concavo-convex, the normals within the lens are 
inclined to the ray on opposite sides of it ; and under the condition 
of mininum deviation these angles are to he equal. Hence 
^3 “ ^4 • 4*3 ~ 4 * 4 : ' '^h ~ '^^4> 
= because the surfaces (2) and (3) are similar curves; (paragraph 
3, above). 
