1888.] Lord McLaren on an Aplanatic Objective. 373 
Let the reciprocals of the focal lengths of the four surfaces he 
denoted by — , — , — , and — . 
'^2 % ^4 
(a) The fourth surface being plane, ^ ^ ^ ^ focal 
length of the plano-concave lens is the same as that of its concave 
surface. 
For the concave surface of the flint lens we have by an elementary 
formula, 
• P2 
(Or by substituting for po value as above) . 
(/^ ~ ^)(^2 ” 1 ' 
_/ yX /^2 ~ ^)(^2 ~ ^) . , . . , ( 19 ). 
• . — 
/^2 
This determines in terms of and the given quantities 
and p.^. 
{h) As the second and third surfaces are identical curves we 
have for the relative refraction of the media, flint and crown, by the 
same elementary formula ^ ^ which determines . 
^ % f.. ' 
Also, v. 2 i Vj, the focal lengths of the surfaces of the crown lens are 
111 
connected by the relation, - = — i — , whence v-, is found in terms 
of and v^. 
From the first surface of the crown lens we determine a-^ in the 
same number as was found : (Eq. 19). 
_ H 
= ( 20 ). 
P'1 
To simplify the notation, the second and third surfaces may now 
be treated as one intermediate surface. Accordingly, in the 
symbols, ^123 4 >i 2 s numerical suffixes apply to the 
first surface, the intermediate surface, and the plane surface, 
respectively. 
(c) To determine $2 in terms of 0^ we observe that after refraction 
