118 
Proceedings of the Royal Society of Edinburgh. 
If P 1 R is given by 
then 
y = px + q, 
p = (A/* + B)/(C/* + D) 
But P 1 R and P 2 R concur on a fixed line 
Hence 
and 
ax + by + & = 0 
a b c 
2 ^2 ^2 
-(C/x + D) q(Cy + D) 
A fx + B 
1/(Z = (C/a + D)F 2 (/x)/F g (/x). 
Thus the equation to P X R may be written as 
A/x + B F o(/x) 
V ~ G/T+D* + ((.>+ l))F,,(yu. 
and OP x is given by 
y = yx. 
= 0 
[Sess. 
• ( 4 ) 
• ( 5 ) 
• ( 6 ) 
(?) 
Put yjx for jul in (7), when we obtain for the locus of P x a quartic with 
a triple point at O r 
Scholium. 
§ 28. In the scholium Maclaurin points out how complicated is the 
task of furnishing a classification of quartics similar to that given by 
Newton for cubic curves. 
He makes it clear that a quartic cannot have more than three double 
points. It seems doubtful whether he was aware that the quartics given 
by Prop. XVII have three double points. 
But he shows that if there are three double points they cannot lie on a 
straight line. 
General Corollary. 
From Props. XIV, XVII, and XIX we conclude that when, in a 
quadrilateral QPjRP.,, the angles at P x and P 2 are constant, while QP X and 
QP S pass through two fixed points O x and 0 2 , then, if any three of the 
vertices lie on given straight lines, the remaining vertex in general 
generates a quartic. 
