374 Proceedings of Royal Soeiety of Edinlurgh. [june 4 , 
at the first surface, the rays converge to a point whose distance 
from the axis of the lens was found to be 
After refraction at the surfaces in contact, the rays converge to 
their pole, whose distance from axis of lens is a^. Neglecting the 
thickness of the lens, and treating arcs as proportional to chords 
(in accordance with the original convention), we have for correlative 
points, — Arc^ - Arc^ : or a ^02 = • 
Hence 
Let X be this ratio ; then 0^ = X6^ . 
(d) To find the ratio of 0^ to 6^ : The rays after refraction at the first 
surface, converge to a point whose distance from axis of lens is Vj. 
After refraction at the plane surface the rays converge to a point 
whose distance from axis of lens is, F . Neglecting the thickness 
of the lens, we find, Arc-^ = Arc^] = ¥6^ \ 6^ = = X'6-^. 
Final Equation of the Achromatic and Aplanatic Surfaces. 
Tracing the ray back from the principal focus. For the plane 
0 
surface (r^ = sec . 6^, we have ?^3 = 1 : oi^ = 0 : cf)^ = 0^ : = — 
f^2 
= <^ 2 ) which has hitherto been considered as an incident ray for 
the crown lens, is, of course, a refracted ray for the flint lens, and 
is equal to cj)^' (by the condition of minimum deviation). 
0 
. <^ 3 ' = ^ which determines the course of the ray in the 
flint lens. 
For the two surfaces of the crown lens, we have 
At the second surface, <f >2 = — ’ 
/^2 
4^2 ~ 4 ^ 2 ' 
At the first surface, = wJ = (n^ + l)^j : cf>-^ = (w^ + 1)— . 
Also, Wj = (7^^ + 1)^1 : <02 = (n^ — 1)^2» ^^4, as before found, 
toi + (02 = 2(^i' = 2(^2 • 
Substituting for these quantities their values, and multiplying 
