361 
1888.] Lord McLaren on an Ajplanatic Objective. 
without making any assumption as to the value of /a, let 2° be the 
corresponding angle of refraction. 
Then under the approximate law ^/<^' = /x, we have /x,= 3°/2'’ 
= 1*5000. Under the true law of refraction we find 
log sin 3” = 8-7188 ; log sin 2“ = 8-5428 ; u= = 1-4996 . 
Sin 2 
The difference between the refractions under the two indices 
/x, = l*5 and jut= 1*4996 is absolutely inappreciable, and it is there- 
fore evident that we may assume = without sensible error, in 
the investigation of the elements of the aplanatic curves. 
The system of curves to be considered is of the form 
. sec [nO ) ; r'^ = a'^ . cos {nO ) ; the first form being derived 
from the second by giving a negative value to n. 
The quantity n may be either integral or fractional, but is always 
greater than unity.* The equation r^ = a]^ . sec (nO) when traced out, 
is found to represent a system of hyperbolic curves which are 
aplanatic for the approximate value of (x under consideration. 
When n = r the curve represented is the equilateral hyperbola. 
I proceed to find a relation between /x and n for refraction at a 
single surface, of the form of the first of the two equations above 
given. 
The axis of symmetry of the curve is the reference line for polar 
coordinates, and is the axis of x for coordinates in x, y. 
r, 0, are polar coordinates of the refracting curve. 
( 0 , is the inclination of the normal to the axis of symmetry. 
0 -, is the angle between radius vector and normal. 
ju,, the index of refraction, is taken at its approximate value . 
o- = ^ =F CO, by a known property of curves whose equations 
are given in this form. Hence 
CO = (?z - 1) 0 for all curves of the form /(sec nO) : and co = (w -h )^, 
for curves of the form, /(cos 0). 
When more than one curved surface is referred to, the quantities 
above mentioned are distinguished by numerical suffixes. 
* When n is less than unity, the refracted ray and radius vector lie on 
the same side of the normal, and the rays do not converge to or diverge from 
the pole. These constitute a family of parabolic or asymptotic curves, which, 
after describing a number of loops depending on the degree of the curve, go 
off to infinity. 
