1888.] Lord McLaren on an A'planatic Ohjective. 363 
If we assume for crown glass of ordinary density the value ju. = 1 *5, 
we find n = — ^ = 3 Or. = 30° and e = sec 30°. 
It will he shown* that the pole is the point of intersection of the 
asymptotes, and is equivalent to the centre of a hyperbola of the 
same eccentricity. 
Comparing these results with those found for the hyperbola, 
p. 359, Eq. 5, it is there shown that for aplanatic convergence to the 
exterior geometrical focus, e= jji; whence if /a be 1*5, the inclination 
of the asymptotes, 0^, will he sec“^(L5) = 29° 36'. 
It is now seen that in the curve /(sec nO), for the same value of 
/A, and for aplanatic convergence to the centre, the angle 0^ must 
he 30°. 
Refraction at two Convex Aplanatic Surfaces of the type 
r^ = a^. sec {nO), {Figure 7.) 
I shall next consider the case of a symmetrical double-convex 
lens, and find the value to be given to n, in order that parallel rays 
may converge to the pole after two refractions. As before, 
nO 
If we neglect the thickness of the lens, and assume Sj and S 2 equi- 
distant from the optical centre, then 0^ = 0i'. 0)2 = 00 ^; also, the angle 
between the normals for corresponding points of the two curves 
is 2o)^ = cji 2 + (fii ; because the exterior angle of the small triangle 
formed by the normals and the intercepted ray is equal to the two 
interior and opposite angles. 
The course of the ray being IS^S 2 P 2 , we have at the first surface — 
L18,N^ = o> = (n-l)e-. 4>,'= 
r 
At the second surface we have, as above, 
nO 
r 
The condition that the refracted ray from Sj shall coincide with the 
refracted ray from S 2 is, as above stated, 2o) = + (f>i . 
Substituting for these quantities their values as above, we find — 
Appendix, p. 377, § 5. 
