114 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Cor. 10. It can also happen when l v l 2 , l 3 are parallel, and a and /3 are 
the angles at which they cut OjOg. 
§ 24. Prop. XVI. 
Let l x and l 3 cut in A. If 0 1 Al 3 = a the curve is a cubic (with a double 
point). 
For 0-,ARP is a cyclic quadrilateral. 
Hence P 1 0 1 R = P 1 AR is constant, so that there is a reduction to Prop. V. 
Similarly, if l 2 and l 3 cut in A 2 and 0 2 A l 3 = /3 the curve is a cubic. 
When both hypotheses hold the curve is a conic, as in Prop. I. 
§ 25. Prop. XVII. 
When in the quadrilateral P x i^P 2 Q it is Q and not R that is restricted 
to lie on a straight line , the locus of R is a quartic curve. 
Bern. 
Let as before 0 1 P 1 and 0 2 P 2 be given by 
L x + AL 2 = 0 . . . . ( 1 ) 
M 1 + /xM 2 = 0 (2) 
Then the condition that Q lies on l 3 leads to a relation 
/x = ( a\ + b)/(c\ + d). 
Hence the equations to P X R and P 2 R may be written in the form 
A 2 L, + AM X + Nj = 0 ..... (3) 
A 2 L 2 + AM 2 + R 2 = 0 ..... (4) 
and the elimination of A from (3) and (4) leads to a quartic equation in x 
and y. 
Cor. 1. The curve does not pass through O x or 0 2 . 
[This description of a quartic is of especial interest. Maclaurin does not 
observe that the curve must possess three double points ; for in virtue 
