106 Proceedings of the Royal Society of Edinburgh. [Sess. 
Hence we may replace P by T and l by l". 
Cor. 5. If 0 1 EB = 0 1 PQ the locus is a conic and not a cubic. For in 
such a case C^PEQ is a cyclic quadrilateral and QO x P is constant, being the 
supplement of PEQ, so that the locus is that of Prop. I. 
§ 17. Prop. XI. 
If, in Prop. X, 0 ± P and 0 2 R simultaneously coincide with 0-fi 2 , then 
the curve degenerates into a conic. 
Cor. 1. Thus, if 
0 2 AB = a 
A0 2 B = /3, 
where B is on V, the locus is a conic. 
Cor. 2. In particular, if a + /3 = tt, and V parallel to AB, the curve is a 
conic, e.g. when a = /3=^_, and V ± r O^. 
The Circular Cubic with a Double Point. 
§ 18. Lemma II. 
This lemma, along with the corollaries attached to it by Maclaurin, 
contains a variety of ways of tracing an important species of cubics 
to which Teixeira has recently drawn attention ( Proc . Ed. Math. 
Society, 1912). 
