1888.] Lord McLaren on an Aplanatic Objective. 
359 
conic sections, the distance of any point in the curve from the focus 
is to its distance from the directrix as e to 1. Or, if the focus he 
origin, and X the distance of the directrix therefrom, ?’ = e(cr + X), 
which agrees with equations 3 and 4. The optical focus is there- 
fore one of the foci of the conic, and it is easy to see that it is the 
further focus. 
If we compare the expression r = e{x + X) with equations 3 and 4, 
it is evident that in equation 3, /x, = e and C = eX , 
and in equation 4, - = e and C = eX . 
In the case of the first figure e is greater than unity, and the 
curve is a hyperbola. 
In the case of the second figure e<l, and the curve is an ellipse. 
When r = C. C is therefore the single ordinate through 
the focus. 
The figures are drawn approximately to scale, for the value fx - f. 
It will be seen from figure (1), that a plano-convex hyperboloid 
having e = /a, is an aplanatic lens for parallel rays coming through 
glass and brought to a focus in air. 
In figure ( 2 ), if a circle be described from centre S cutting the 
ellipse, the convexo-concave lens will bring the parallel pencil 
accurately to a focus at S. 
Figures 3 and 4 are intended to show the course of converging 
and diverging pencils for concave hyberbolic and elliptic surfaces. 
Xumbers are attached to the angles showing the relative values of 
sin cf> and sin for /x = f . 
Figure 3 represents a section of a concave hyperbolic surface, glass 
to the left, air to the right. Now, if the medium to the left were air, 
it has been seen that parallel rays coming from the right would 
converge to a focus at F, the exterior focus of the hyperbola. But 
as the medium to the left is glass, a converging pencil making angle 
PiSiNi with the normal will be refracted through the smaller angle 
128 ^X 2 , and rendered parallel to the axis. 
Figure 4 represents a concave elliptical surface, glass to the right. 
If the medium to the right were air, it has been seen, that parallel 
rays coming from the right would converge to a focus at F. But 
as the medium to the right is glass, a pencil converging to the focus 
at F, and making the angle PSN^ with the normal will be refracted 
