376 
Proceedings of Royal Soeiety of Edinburgh, [june 4, 
the curvature of objectives for any values of and Because 
the difference between the actual and the mean values of and 
could only affect the eccentricity of the aplanatic curve in a very 
small degree, and the difference between the ordinates given in the 
table, and those appropriate to the actual values of and would 
be of the second order of small quantities, in comparison with the 
difference between either of these and the ordinates of circular 
curvature. 
I conclude by pointing out what I conceive to be the essential 
feature of the preceding investigation, which consists in the sub- 
stitution of the approximate ratio </>/<^' in place of the true ratio of 
refraction ^ , and thereby determining the elements of the surfaces 
of an achromatic combination which shall be also sensibly aplanatic. 
It is evidently impossible to compute directly the values of e for a 
series of surfaces in combination, consistently with the true ratio 
sin 
To do so would involve the solution of equations between 
the sines of sums and differences of six variable angles, <^2 * 5^3 
^ 1 ' (f >2 ^ 3 ^ and the sines of sums and differences of six other variable 
angles, O .2 0^ tOg Wg. The approximate solution here indicated 
may be regarded as true within the limits of errors of workmanship. 
Appendix A. 
Note on Properties of the Auxiliary Curves. 
1. Every curve of the class first considered, /(sec ^), is of the 
general form of a hyperbola ; that is to say, it is a continuous curve 
extending in two branches to infinity, and having neither nodes nor 
points of inflexion. This statement may easily be verified by deter- 
mining the radii for a few values of 0, when the law of the curves 
will become evident. The condition must be observed that n>l. 
If <1, the surface taken singly will not bring rays to a focus, and 
the plane curve has loops depending on the degree. 
2. Every such curve has two real asymptotes, intersecting in a centre 
and their inclination to the axis is found directly by putting r = go , 
which gives for 6 the value — 
2n- 
3. If we write the equation in the form r = a . sec l(nO), 
