378 Proceedings of Royal Soeiety of Pdinlurgli. [june 4 , 
7. It appears that the centre of the system which has been taken 
as pole, possesses some of the properties of a focus, if we give to the 
term focus the extended signification of a point from which radii 
make angles with the normal in a fixed ratio to the angles made by 
radii from another focus. In this case the corresponding focus is 
evidently a point at infinity. 
But further, in the limiting form of the equilateral hyperbola, 
there are the two interior foci of the ordinary hyperbola, and it is 
therefore probable that the curves of the polar degrees also 
have interior foci ; but these have not been investigated. 
8. By considering the relative positions of the normals of a curve 
of the polar degree, and the hyperbola having the same trans- 
verse axis, centre, vertex, and asymptotes, it is seen that the first- 
mentioned curve lies within the hyperbola, which it touches only at 
the vertex and at infinity, except in the case of = 2, where, as 
already pointed out, the curves are identical. The nearer the index 
is to 2, the less difference there will be between the polar curve 
and the corresponding hyperbola. 
Appendix B. 
Compound Ohjective^ Flint Lens at the Front. 
A very simple expression for this combination is obtained by 
considering the rays as being cut orthogonally by a spherical 
interior surface ; so that there are only two refractions to be con- 
sidered, namely, those of the two exterior surfaces. Although the 
assumption that the rays are cut orthogonally is incorrect in fact, 
yet, as the relative refraction of flint and crown glass is small, the 
deflection of the rays at the interior surface may in this case be 
neglected. 
As there is a superfluous condition, I shall assume equal values of 
n for the exterior surfaces of the crown and flint lenses, and find 
the values to be given to their parameters, a-pi. 2 , so that the rays 
may pass through the compound lens at the angle of minimum 
deviation, reckoned from the two exterior surfaces. The 1st surface 
is of the form, f{coBnO), and the 2nd surface is of the form, 
/(sec n$). 
Considering the arcs of the lens-surfaces as proportional to their 
sines (in accordance with the original convention), we have — 
