286 Proceedings of Royal Society of Edinburgh. 
(4) From ( b ), (c'), and ( d ') we obtain similarly 
[sess. 
r = a sec m v ; p = a • sec ™ ^ • v = a • sec 
(«') 
<i(a • sec m v) 
By a known relation between r, p, and tbe radius of curvature p , 
p = i 
dr 
dp 
p r dr 
= 2 dip = a * C0S m v = v (^ n case )* • (6) 
Similarly, £ = a • sec m + V = v (also in the second case). ... (6) 
On this property of the system of curves (which appears to have 
been hitherto unnoticed) the determination of the equation of tbe 
caustic is founded. 
The property is this : — The semi-radius of curvature of any curve 
of the system is equal in length to the radius-vector of the negative- 
pedal, or next higher curve of the system ; so that, if we distinguish 
these radii by suffixes corresponding to the degrees of the respective 
curves we have 
From the two equations (6) we have also, = a • sec m v • sec v 
A 
= r sec v , where p and r are the radius of curvature and radius- 
vector of the same curve (6a) 
Moreover, since v ■= m6 we have for all curves of the series 
<o = (m±l)0; if/— 1)0 , where the signs are determined by the 
geometrical construction 
3. Reflexion-Caustics of Two-Term Polar Curves of the Degree , 
Case 1. — Equation of the Reflexion-Caustic of the Curve, 
T ± 0 
t m — a m sec — • 
m 
In this case, the primitive curve is concave to the pole. 
The accompanying diagram (fig. 2) is drawn to scale for curves of 
