1889-90.] Lord McLaren on Reflexion-Caustics of Curves. 293 
extension of this theorem to all two-term polar equations has been 
previously given. 
By means of Formula 10, the equations of the following curves 
with their caustics may be written : — 
Reflecting Curve. Caustic. 
m = 
2 
r = 
a ■ 
■sec 
2 (i 
0) 
R = 
= 0 
• (Parabola). 
m = 
3 
r = 
a- 
sec 
3 U 
6) 
R = 
= 
sec 
4 (i ey 
■sin(i 
6). 
m = 
4 
r = 
a ' 
■sec 
4 (i 
6) 
R = 
= a- 
sec 
*{^ey 
• sin ( f 
e). 
m — 
5 
r = 
a- 
sec 
5 (i 
0) 
R = 
= a 
•sec 
•sin( | 
6). 
m = 
8 
r = 
ci- 
■ sec 
8 (l 
6) 
R = 
= a- 
•sec 
9 (tW 
sin(f 
6). 
m = 
11 
r = 
a< 
•sec 
11 /JL 
Vi 1 
0) 
R = 
= a 
•sec 
• sin ( f- 
6). 
m — 
14 
r = 
a> 
• sec 
14/ 1 
V 1 4 
0) 
R = 
= a 
•sec 
15 (A-tf) 
• sin(|- 
6 ). 
m — 
17 
r = 
a - 
• sec 
17 /_!_ 
Vl 7 
6) 
R = 
= a- 
•sec 
18 (*0) 
■sin(A 
6). 
m = 
20 
r = 
a 
•sec 
20/ 1 
\20 
6) 
R = 
= a • 
•sec 
“(A*)- 
sin (A 
6). 
For all values of m exceeding 5, the coefficients of 6 follow a 
regular law. If m = Zn + 5, the first coefficient of 6 is — - — , 
6rc + 9 
• w -f- 1 
and the second coefficient of 0 is — • 
2n + 3 
Similar results might be given for the caustics comprised in 
Formula 12, and of course in both cases, also for fractional indices 
of the secants corresponding to integer indices of r and a in the 
usual form of the equation. 
Case 6. — Caustics of Oblique Pencils. 
The case of a pencil of parallel rays inclined to the axis of the 
reflecting curve may be next considered. 
Let (3 be the inclination of the pencil to the axis of symmetry, 
then the inclination of the reflected ray to the normal is w + /?, and 
the distance of the new focal point, C', is i - • cos (to + / 3 ) . 
2 
By inspection of diagram 2, it is evident that as OC'S is a right 
angle, the new point C' lies on the circle 3SOQ, but is nearer the 
reflecting curve or further from it, according as the incident ray is 
inclined from the axis or towards it. 
In going over the proof, if we allow for the quantity (3, we shall 
find for 0 in the caustic the value (2m-l)v±/3; and for R the 
value r' sin (m -2 )v± (3. 
