1888 .] Lord McLaren on an Aplanatic Objective. 
357 
chief object of this paper. But before entering on this i3art of the 
subject, it seems desirable to consider whether the prescribed 
conditions can be wholly or partially fulfilled by curves of the 
second degree; because, for certain values of the constants, the 
aplanatic curve takes the limiting form of a circle, ellipse, or hyper- 
bola, and it will be shown that in such cases the refractive index, 
«, = either e or - . 
^ e 
Aplanatic Circles . — The general equation of an aplanatic curve 
being as stated, + c, when the absolute term is wanting, this 
reduces to a circle, where is a ray emanating from an 
exterior focus, and converging to or diverging from a focus within 
the circle. The equation may also be written sin $2 = p sin 0^, where 
and $2 the inclinations of the ray to the line joining the two 
foci. These angles, it appears, are equal to the angles of refraction 
and incidence respectively, oi $2 = 4* ’ ^i — the ordinary notation. 
For any point within the circle which may be taken as pole, there is 
a corresponding pole exterior to the circle, for which the given 
relation holds. The condition that the coefficient /x shall be equal 
to the refractive index of the glass, determines the position of the 
foci or poles relative to the centre of the circle. The distance of 
the foci is evidently the sum of the least values of and and 
the difference between the greatest and least values of the exterior 
radial coordinate is equal to the diameter of the circle. From these 
elements and the given refractive index, the position of the poles 
may be directly found. 
Aplanatic Cukves of the Second Degree. 
An ellipse or a hyperbola is only aplanatic for parallel rays. An 
elliptic surface brings rays that are parallel in air to a focus in glass. 
A hyperbolic surface brings rays that are parallel in glass to a focus 
in air. A plano-convex hyperbolic lens is therefore aplanatic in the 
strict sense; and it will be shown (by the method of approximate 
curves) that a double convex hyperbolic lens is sensibly though not 
strictly aplanatic. It is, in fact, the type of the figured crown- 
glass objective. 
I shall first show that u = e in the aplanatic hyperbola, and p = ~ 
in the aplanatic ellipse, for refraction at a single surface (figures 1 
