1889-90.] Lord McLaren on Reflection-Caustics of Curves. 287 
the 4th and 5th fractional degrees of r and a ; but is applicable to 
the proof for polar equations of any fractional degree. 
OV is the reference line and axis of symmetry of the curves. 
SS' is the primary, or reflecting curve, which may be denoted by 
</> 1 ; SS', its negative pedal, or </> 2 . Their equations may be written 
for present purposes in the form 
r — a • sec m (—) = a sec m v : 
\m' 
v=a- sec m ( ) = a sec m Vm . . . (d') 
\m+ V 
S is the incident point, and S a corresponding point on the negative- 
pedal, to which radii OS, OS are drawn. 
SC is the reflected ray. 
OP is a perpendicular on the tangent to the primary at S, or 
radius-vector of its pedal. 
The rectangular figure OSSQ' being completed, let a circle be 
described from T, the intersection of diagonals, as cenrte through the 
four points 0, S, S, Q j and let QC be drawn perpendicular to the 
reflected ray, and therefore meeting it at a point C on the circle. 
(1) As the figure SSOQ is rectangular by construction, ^SOS = 
^:QSO; SOS is also analytically = ^SOP = v, because they are 
respectively the angles between radius-vector and perpendicular on 
tangent of the two curves, <f> 2 and <£ 15 of the system (by 5). 
. *. ^iQSO = <^.SOP ; and SQ is parallel to OP the perpendicular 
on the tangent to SS' at S. 
.*. SQ is a normal to SS' at S, and being also = SO or v, it is the 
semi-radius of curvature (by 6). 
Also SC = SQ- cos QSC= £ • cos i = £-• cos w . . . (7) 
A A 
.*. C is a point on the caustic (by 2). 
(2) To find its coordinates E and 0, we have 
0 = ^SOV - ^ SOC = ^SOV - ^ SQC = 0 - (90° - w ), 
since the points Q and 0 lie on the circumference of a circle ; or, 
by expressing all the angles in terms of v, 
.*. 0 = (2m - l)v - — j 
