98 
Proceedings of the Royal Society of Edinburgh. [Sess. 
There is thus a projective correspondence between the ray 0 2 Q and the 
tangent PQ to the conic, and Q generates the singular cubic. For the 
present he is restricted to the use of straight lines as loci, and of these he 
uses two.] 
§ 11. Projp. VI 
shows how to determine the asymptotes, and also the species of the cubic, 
according to Newton’s classification of cubics. 
The next theorem is Lemma I. 
If 0 is a fixed point, P any point on a - given straight line , and OPQ a 
triangle of given species, then the locus of Q is a straight line. 
We need not add the proof. 
§ 12. Prop. VII. 
All the cubics of Prop. V may be obtained by taking L QOJd — Tr. 
Find K on V such that O^K = /3. Let K0 1 0 2 = y. Draw OfT so that 
P0 1 T = 7r — y, and let it meet QP in T and Q0 2 in S. Then, by the lemma, 
when P moves on l, T generates a straight line, while S also generates a 
straight line l" (by Prop. III). 
We may thus obtain the locus of Q from the constant angle STQ and 
the intersection of 0 2 S with QT. 
Cor. 1. Either a or /3 may be replaced by a right angle or by an angle 
of any given magnitude. 
This is easily deduced by starting from TQS. 
The remaining cor. discuss the asymptotes and a variety of particular 
cases. 
E.g. Maclaurin notes that when 0 2 goes to infinity, the pencil of lines 
becomes a system of parallel lines. Special cases arise when l and V are 
parallel, or when the rays are inclined to l' at angle a. 
§ 13. Prop. VIII 
considers the reduction of the equation of the cubic to a standard 
Newtonian form. 
Some particular sub- cases are given. 
Ex. 1. 
7 r 
Let l and V be parallel and perpendicular to 0 1 0 2 , and let 0 1 PQ = " 2 ’. 
Choose the origin at 0 2 , and let A, B, O x be the points ( a , o), (b, o), (d, o). 
