292 Proceedings of Royal Society of Edinburgh. [sess. 
(4) Comparing (4) with (2) w~e see that the ratios } are 
V l V 2 
equal. But, as before found, see Title (2), Equation (6), v 1 = ; 
finally, v 2 = ~p 2 (5) 
It is thus proved that for all curves of the form, 
r n = a n cos mO , 
where the exponent may be either integral or fractional, the semi- 
radius of curvature of the primary curve is equal to the radius-vector 
of the negative-pedal.* 
(5) It results from the value last found that all the geometrical 
relations deduced from the diagrams for the fundamental curves 
are true for similarly constructed figures applicable to the curves 
here considered, where m and n have different values. 
If we take a system of curves in the form of the figure of 
diagram 4, we find for B 2 and 0 2 — 
Erom the relation, 
^iSOC' = ^SQ'C' = 90° - QSC = 90° - <o 2 ; 
® 2 = 0 2 + SOC' = 0 2 + f -« 2 ; .... (6) 
Erom the relation, ^.CO^S = - SOC' = v 2 + « 2 - , 
.*. R 2 = 0^5 • cos C(E2 =u 2 • sin (v + a> 2 ) .... (7) 
From these simultaneous equations, a point R 2 , 0 2 on the caustic 
may be found corresponding to any point on the primary ; and from 
(3) and (4), points on the positive and negative pedals may be 
similarly determined. 
Lastly , It is evident, by inspection of the diagrams, that all 
circles described on a semi-radius of curvature, as diameter, pass 
through the pole, which pole is either a centre or (in the case of 
parabolic curves) a focus. 
This curious geometrical property is known to be a property of 
parabolas of the 2nd degree, and from it, if four points be given, 
the focus of the parabola may be found. I am not aware that the 
* The proof probably admits of extension to the form r n = a n cosmO ■ cos jpd, 
but this has not been completely investigated. 
