360 Proceedings of Royal Society of Edinhurgh. [june 4 , 
through the greater angle 128^1^25 rendered parallel to the 
axis. 
Figure 5 is intended to show how an aplanatic combination may 
be made from an elliptic and two hyperbolic surfaces. 
The course of the ray is as follows : — the parallel pencil Is, is 
refracted at the elliptic surface s, (^eccentricity = —\ and the 
\ 
rays converge towards the focus F;^ 
By refraction at the hyperbolic surface ss^ 
the rays are 
rendered parallel ; and by a third refraction at the hyperbolic surface 
ssfe = /X.2) they are made to converge accurately to F2, the focus of 
ss^ and principal focus of the telescope. 
Such a combination, however, would be of little value for optical 
purposes; because the rays would not be inclined to the lens- 
surfaces at the angle of minimum deviation, and would therefore 
not satisfy one of the conditions for the correction of chromatic 
error. I proceed to show how curves which satisfy the conditions 
of achromatism may be found. 
Appeoximate Aplanatic Curves. 
If instead of the true law of refraction, = u, we assume (for 
’ sin ^ 
the purpose of determining the curves of aplanatism) an approximate 
law <^/<^' = /x, we obtain a system of approximately aplanatic curves 
the elements of which can be determined with great facility for any 
required combination. 
Now, when it is considered that the spherical arc of an object 
glass, from centre to boundary, cannot exceed from 2 ° to 3 °, it is 
evident that the difference in such a lens between the ratio and 
sin 
the ratio is extremely small. Accordingly, a surface which 
is aplanatic for the approximate law, = will be sensibly 
sin <(> 
aplanatic for the true law of refraction ^ within the limits of 
^ sin <j6 
the aperture of an ordinary lens. 
If, for example, the lens have an arc of curvature of 6°, then for 
a parallel pencil 3 ° is the greatest possible incident angle ; and 
