1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 115 
of (3) and (4) it must be a unicursal curve, and the double points are 
given by 
Li/L 2 = Mj/M 2 = Nj/Ng (5) 
(vide “Courbes Unicursales,” L’Ens. Math., 1912). 
The equations (3) and (4) are not the most general of their kind, for the 
envelope is in each case a parabola. But it may be shown that any 
unicursal quartic with three double points may be considered generated by 
the intersection of two lines, 
LA 2 +M 1 Xh-N 1 =0 
L 2 A. 2 + M 2 A. + R 2 == 
which envelop two conics, and which may be obtained by making a constant 
angle 0 1 P 1 R move with its vertex P 1 on a circle (or a straight line), and 
similarly a constant angle 0 2 P 2 R move with its vertex P 2 on another circle 
(or straight line), while Q lies on a conic through O x and 0 2 . We may 
show as before that, without loss of generalisation, this conic may be 
replaced by a straight line. Maclaurin’s generation is therefore the 
simplest of the above, and it is an easy step to proceed from it to the more 
general one in which circles are employed. It must not be forgotten that 
in Part I he only makes use of linear loci.] 
§ 26. Prop. XVIII. 
If in Prop. XVII l v l 2 , l B are parallel, the curve is a cubic. 
For the parabolic envelopes have in common the tangent line at infinity, 
so that the quartic reduces to this line and a cubic. 
Particular Cases. 
(More generally we find a cubic when two corresponding tangents to 
the parabolas coincide.) 
Let « = = ^ ; and let l v l 2 , l 3 be X'O-jOg. 
In the figure choose 0 1 0 2 as cc-axis with origin at 0. 
Let OO^a; 00 2 = 6; OD^cZ; 0D 2 = <$; OQ = y. 
