1889 - 90 .] Lord McLaren on Reflexion-Caustics of Curves. 289 
But these formulae are not required for the proof, which, as in the 
preceding case, is purely geometrical. 
(2) To determine the coordinates of R and 0, we have 
© = ^SOV - ^SOC = ^SOV - ^SQC = 0 - (90° - ©). 
0 = (2m + l)v - ^ , 
or by changing the reference line to Y ; 0 = (2m + l)v. . . (11) 
(3) Also, 
R = OC = 02- sin 02C = 02 -sin OSC ; 
= 02 • sin (QSC + QSO) = Ot • sin (QSC + S02) ; 
.*. R = v • sin (w + v) = a • cos m_1 i/ • sin(w + l)v + v 
= a • cos TO_1 i/ • sin (m + 2) v . 
(by taking the value of v from (e). 
Substituting for v its value from (11), v=( -~® , V we have 
& x ' \2m+ 1/ 
finally — 
B = a ■ cos m ~ 1 (' ® V sin( m + ^ o) , or 
\2m + 1 / '2m +1 / 
R-sec™ Y — ) = a sin( ^ - ~ 2 o) 
\ 2m +1/ '2m +1 / 
. ( 12 ) 
I It is easily seen that Cases 1 and 2 are closely related. The 
curve which passes through corresponding points of the series of 
pedals is a spiral which never attains the centre, having an infinite 
number of convolutions in both directions. 
! 
Cases 3 and 4. — Reflexion-Caustics of Curves which are Convex 
to the Pole. 
The equations of these curves are of the forms — 
r n = a n • sec nO ; 
r n = a n • cos n6. 
In the curves whose equations are in this paper considered, the 
pole is evidently either a focus or a centre. In the equations with 
fractional exponents which have been already considered, the curve 
is generally parabolic and the pole is an interior focus. But if the 
VOL. XVII. 25/9/90 T 
