294 
Proceedings of Boyal Society of Edinburgh. 
b 
The equation of the caustic of oblique pencils (Case 1) is accord- 
ingly 
R = a sec 
m+ i / p±P \ 
\2m-l) 
sin 
(m- 2)9 + /? 
(2m -1) 
(13). 
In the parabola we have from this equation 
B - a - e c(! + s) 
3 
smv; or R = a-sinysec 3 ^ . 
By taking a new reference line, making ^- = y with the old, this 
reduces to 
e 
R = sec 3 
3 ’ 
which is the equation found by an independent geometrical proof 
in the paper read to the Royal Astronomical Society, referred to in 
the introductory paragraph of this paper. 
We now see that this is only a particular case of the form (12) of 
the general equation of the reflexion-caustics of polar curves. 
4. Locus of the Field of View in the Optical Caustic. 
In the preceding paragraph it is pointed out that for a given point 
S on the reflecting surface, all the focal points C, C', &c., for pencils 
of different inclinations lie on the same circle ^SOQ. 
If S 0 be the vertex of the reflecting curve, the series of points C, C', 
&c., are the principal foci for different pencils (e.g., stars in the 
telescopic field), and these all lie on a circle in a principal plane 
whose diameter is the principal focal length. For a reflecting sur- 
face of revolution the field of view is accordingly the surface of a 
sphere whose diameter is the focal length. 
From a note communicated to me by Professor Cayley, and 
quoted in the above-mentioned paper, it would appear that this is a 
property of all reflecting surfaces. It may now be added that if 
any other point Sj on the reflecting surface be taken, the foci for 
all pencils reflected at this point in a principal plane also lie on a 
circle whose diameter is half the radius of curvature at the point. 
This property is not confined to the polar curves here investigated, 
because it results from the general equation for the length of the 
reflected ray as given by Sir William Hamilton in the paper above 
referred to. This circle corresponds to the generating circle of the 
