1889-90.] Lord M/Laren on ’Reflexion-Caustics of Curves. 285 
This is a property of plane curves in general, and is not peculiar 
to the systems of curves which are considered in this paper. 
By a known property of polar equations of the forms — 
r m = a m cos mO , . . (a), and r m = a m sec mO , . . (6), 
we have v = m6 for all values of ra, integral or fractional. This may 
he verified by logarithmic differentiation of the equations. The 
forms (a) and ( b ) are the generalised forms of the equations of the 
primary curves whose reflexion-caustics are to be found. 
(1) From ( a ) the first form of the equation we have 
™ra+ 1 m — — 
7 • m+1 
p> — r cos v — — ^ r m+ = a m p ; or, r = a m p 
a 
Substituting for r its value, the equation of the pedal is 
m m 
^? m + 1 = a m+1 cosv; (c) 
and similarly, we find for the equation of the negative-pedal, 
m m 
v 1_m = a 1_m cos (d) 
(2) From (b) the second form of the equation of the primary, we 
find for the equation of the pedal, 
m in 
= a~ m sec v ; 
and for the equation of the negative-pedal, 
m m 
v . m+i — : a m+1 sec v 
(0 
(d') 
(3) In order to find a geometrical relation from the polar differen- 
tial expressions, let the quantities r, p, v , and a be reduced to the 
first power, then from (a), (c), and (d) we have, by reduction and 
differentiation, 
■y 1 _1 j 
r = a-cos™v; p = a-cos m (v); v = a- cos m (v). . 
3_ J_ +1 
dr d(a • cos m v) d( sec m v) d(sec 2 v) 0nnnmM 
= U S6C V e • » 
dp J_ + 1 J- 
d(a • cos m v ) d ( sec m v) d(sec v) 
(«) 
(/) 
