364 Proceedings of Royal Society of Ediiiburgli. [june 4 , 
(2n-\)e 
{2n - 2)9 = — — , whence 
_2n-\ 2^-1 /"2/x — 2'^ 
( 7 ). 
The eccentricity, e, is the secant of this angle. 
For the value />t = 1*5 this reduces to n = 2 and ^q = 45°. 
To verify the equation, we have for the same value of fx, 
= ^0 : <f >2 = ^9' : also w = ^, whence + cfi^ =29 = 2w, which 
satisfies the condition that the two interior rays shall coincide. 
For the values /x = 1 *5 and n = 2, the equation of the curve is 
P = a^. sec {29) : or ?’2cos (2^) = a^^, . . . (8), 
the polar equation of the equilateral hyperbola from the centre. 
This is the figure of a single aplanatic symmetrical lens for bringing- 
parallel rays to a focus at the pole of the second surface. 
Refraction at a single Aplanatic Surface of the type f (cos n9), 
{Figure 8.) 
The equation of the curve is, r” = a^cos {n9)^ and by its known 
properties, the angle between radius vector and normal is n9. 
Hence w = (?^+l) 9, and the focus is on the concave side of the 
curve. 
The curve / (sec 9) was found to be the analogue of the hyperbola, 
in this respect, that it brings rays which are parallel in glass to a 
focus in air, on the convex side of the curve. The curve, / (cos ^), 
is the analogue of the ellipse, as it brings rays which are parallel in 
air to a focus in glass, and is concave to the optical focus. 
The figure gives by inspection the values 
(^ = (0 = (?^ + 1)^. = whence {n^-V)9 = pn9. 
fX = 
n+\ 
n 
(9). 
For the value /x = 3/2, the value of n is 2, and the equation is 
that of the Lemniscate, = a? cos 29 (1^)- 
If the surface considered be concave, this is of course equivalent 
to a convex surface of air. The new p is the reciprocal of p in the 
convex surface, and the relation n = 
gives a negative value of 
n. Substituting this negative value in the equation we find 
