362 Proceedings of Poyal Society of EdinlurgJi. [june 4, 
Refraction at One Aplanatic Surface of the type r” = a” mc{nO). 
{Figure 6.) 
In the family of curves under consideration, the pole is on the 
convex side of the curve, and corresponds, as will he shown, to the 
centre of a hyperbola. It is geometrically evident that the radius- 
vector and incident ray lie on opposite sides of the normal, and the 
condition of aplanatism is that the refracted ray shall coincide with 
the radius vector, — the pole of the curve being then a true 
optical focus. In figure 6, the parallel pencil passes through glass 
and is brought by refraction at the surface to a focus in air. 
The equation of the curve being, as stated, r^ = a^ . sec we are 
to find (1) the value which must be given to n in terms of /x,, so 
that the parallel pencil may converge to the pole under the 
approximate law <^/^' = /x, ; and (2) the inclination of the asymptotes 
to the axis 0 ^ ; and the eccentricity, sec 0^^ ^ Tracing the 
course of the ray backwards from the pole, we have at any point 
S on the refracting surface (figure 6). 
ZNSP =</) =o■ = 7^^ 
nO 
L 1ST =<f> = — (by the assumed approximate law). 
A*' 
The condition that the rays coming from P shall be brought by 
refraction into parallelism with the axis, evidently is that <f>' is to 
be equal to a> ; or. 
<fi ={n- 1)0 , whence, nO = yi{n -1)0 
^ = 
n-1 ' /X, - 1 ’ 
JU, fl 
and the equation of the curve is r^^~^ = a^~^ . sec^ ‘ ‘ 
The inclination of asymptotes to axis is, 0q = ^ = -tt . 
0 2n 2/x, 
The eccentricity e = sec^ = sec . 
This is the value found by putting r= oo . See pp. 376-7, paragraphs 2 and 6. 
