112 Proceedings of the Koyal Society of Edinburgh. [Sess. 
Ex. 5. O x PQ — ; V and l J_ r 0 1 0 2 : Q0 2 R three collinear points. 
Equation xy 2 (b — d) = (x — d)(ay 2 + a — bx 2 ). 
§ 21. Projp. XIII. 
When Q and R move on fixed straight lines V and l", then the locus of 
P is in general a cubic with a double point at 0 V 
Maclaurin’s proof is analytic. 
The geometrical method he would employ later runs thus : — 
Leave Q free, and restrict P to lie on a straight line m. Then Q lies on 
a cubic cutting V in three points Q x , Q 2 , Q s , to which correspond P 1? P 2 , P 3 on 
m. Hence, when Q lies on V, P traces out a curve cut by m in three points 
P 15 P 2 , P 3 . The curve is therefore in general a cubic. 
But it may degenerate. 
SECTION III. 
On the Description of Lines of the Fourth Order, and those 
of the Third Order which have no Double Point. 
§ 22. “We have described Lines of the Second Order by the rotation of 
two constant angles round two fixed points ; also Lines of the Third Or^ler 
by the use of as many angles, of which we have supposed one to be rotated 
round a fixed point, while the other is conducted along a fixed straight line. 
“We now proceed to the description of Lines of the Fourth Order by 
conducting each angle along a straight line.” (The quartics obtained have 
either two or three double points.) 
Prop. XIV. 
Given 0 ± and 0 2 two fixed points ; L 0 1 P 1 i2 = a ; L 0 2 P 2 R = /3, constant 
angles , where P x and P 2 lies on fixed lines l x and l 2 respectively. 
