288 Proceedings of Boyal Society of Edinburgh. [sess. 
or again, by changing the reference line for the caustic from X to Y, 
0 = (2m - l)j/ (8) 
(3) Also, R = 0C = 02 • sin 02C = 02 • sin OSC 
= 02 • sin (QSC - QSO) ; = 02 • sin (QSC - S02) ; 
. \ R = v • sin (a) - v) = a • sec m+ V • sin (m-2)v (9) 
(by taking the value of v from (e').* 
If, in this equation, we substitute for v its value from (8), ; 
0 
2m - 1 5 
we obtain finally, 
R = a • sec 
m+ 1 
e 
(2m - 1) 
e 
sin 
m 
2m 
or 
R . cos w+1 - = a sin ( ^— 2 ©) ; 
2m - 1 v 2m - 1 / 
( 10 ) 
the equation of the Reflexion-Caustic. 
From (9) we see that when v = 0, R = 0, and there is a cusp at 
the principal focus. 
When v = 
m - 2 ’ 2 
, R = r, and the caustic meets the reflecting 
curve. 
When v = 
R is infinite. 
Case 2. — Reflexion-Caustic of the Curve , 
-)• 
■ m/ 
JL p 
r m — a m cos 
The primitive curve in this case also is concave to the pole. 
(1) In diagram (3) the construction is similar to diagram (2), only 
the form of the curve is different, and the order of the points 2, S 
and P is reversed. The proof is substantially the same as in case 
1, as far as equation (7), and C is a point on the caustic. 
By ( c ) and ( d ) the equation of the pedal is given by changing m 
into m , , and that of the negative-pedal by changing m into m 
m + 1 
1 - m 
* The change of the reference line does not affect the value of R, because the 
coefficient, sin (co - v) is a function of the difference of two angles. 
