290 Proceedings of Royal Society of Edinburgh. 
exponent, n, is a whole number, the curve is always central (as will 
be shown in the sequel), and it consists either of infinite branches 
which are convex to the centre, or of loops or “ foliations ” passing 
through the centre. These forms are shown in diagram (4), which 
has been very carefully drawn to illustrate the proof of the general 
theorem for the case of curves of integral degrees and integral 
coefficients of 0. 
The curve SS'S' is a branch of a real curve of the 4th degree 
(copied from my paper on “ Homogeneous Equations ”), consisting 
of four hyperbolic branches symmetrically placed about 0 as a 
centre. The four foliations of the pedal pass through the centre 0, 
which is a point of inflexion for these. Only two of the foliations, 
P, P', are shown in the diagram. The negative pedal, 3^', is neces- 
sarily parabolic, because only at the point infinity does its tangent 
become perpendicular to the radius-vector, or asymptote, of the 
primary curve. 
In the figure, S is the incident point. 
S, 2, and P are corresponding points in the three curves, from 
which radii are drawn to O. 
SQ is the normal produced to SQ' in the opposite direction.* 
The values of R and 0 are found as before, attention being paid 
to the signs in the general expressions (6 b) viz. a> = (ra±l)0; 
i]/=(m+ 1) 6. 
To avoid fractional expressions it is convenient to reduce angles to 
6 instead of v. It is unnecessary to give a figure for each variety. 
For curves of the forms shown in figure (4), the equation of 
the primary being r n = a n cos (nO), that of the caustic is 
Case 5. — Reflexion-Caustic of the Curve r n = a n ’ cos m6 . 
Where the coefficient of 6 has a value different from the exponent 
of r and a , we can also obtain geometrical solutions or simultaneous 
* SC is the reflected ray produced to C' in the opposite direction. The 
proof is the same as in Case 1 , as far as Equation (7), if we substitute SC' 
for SC. Then taking SC = SC', C is a point on the caustic. It is now 
evident that the curve should have been treated as a convex reflector. C' is 
a point on the true caustic, if the direction of the incident ray be reversed. 
R = a • cos 71 
1 -n 
