1888.] Lord McLaren on an A^lanatic Ohjective. 365 
r"” = a"”cos which is equivalent to r” = a”sec??^. Hence, for 
a concave surface, it is the curve, / (sec nO)^ which brings rays to a 
focus in glass ; similarly, for a concave surface, the curve, / (cos nO), 
will produce convergence to a focus in air. 
A double convex aplanatic lens cannot have both its surfaces of 
the type, / (cos nO) ; because the poles lie to the concave sides of the 
curves, and there is no exterior focus. 
It is evident that the proper form for a double convex lens which 
is to enter into an achromatic combination, is the form in which the 
two species of the equation are used, the surface, / (cos nO)^ being 
towards the pencil of parallel rays, and the surface, / (sec n^O), being 
towards the principal focus, in accordance with their respective 
properties. The equations of the surfaces of such a lens will now 
be found, and they will afterwards be applied with proper values of 
IX and a to the case of an achromatic and aplanatic combination. 
Refraction at the Two Surfaces /(cos n-fi) and /(sec nfd) of a 
Double Convex Lens. 
(1) I will first suppose the factors and n^, as well as the 
parameters a-^ and a^ to be equal. 
As the rays incident on the first surface are parallel rays, the 
first surface is of the form, /(cosw^^): .*. coj = (72,-f l)^j. As the 
convergence is to an exterior focus, the second surface is of the form, 
/(sec nO ^ ; and oi^ = (n- 1)^2* 
The radii of the two curves are to the same side, i.e., towards 
a common focus, if we neglect the thickness of the lens; and for 
correlative points on the two surfaces, = B^^ 
At the first surface, = (w + 1)^ ; = {n+\)- . 
A 
At the second surface, cf>2 =nB; . 
fX 
The condition of aplanatism is that the rays refracted into 
the lens from the two surfaces coincide; or, that their inclina- 
tion to the axis is the same. Inclination of ray 8^82 to axis 
is at the first surface, = ; and 
at the second surface = ~ ^2’ whence, by equating these expressions 
for the inclination of the ray within the lens, — Wj + o>2 = </>/ -f cf)^ . 
8ubstituting for these quantities their values, as above. 
