1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 113 
If R is restricted to lie on a straight line l 3 , the intersection Q of 0 1 P 1 
and 0 2 P 2 in general generates a quartic having double points at O x and 0 2 . 
Dem. 
Let 0 1 P 1 have equation 
and 0 2 P 2 have equation 
L 1 + XL 2 = 0 
M x + />tM 2 = 0 
Then P 1 R has an equation of the form 
X 2 N 1 + XN 2 + N 3 = 0, 
or 
xk 2 {\) + yty X) + C 2 (X) = 0 . 
( 1 ) 
( 2 ) 
(3) 
Similarly, P 2 R has an equation of the form 
x fM + + i'M = 0 G) 
The condition that R lies on the line l 3 , viz. on 
gives rise to the condition 
lx + my + n — 0 . 
I m n 
A 2 (A) B 2 (A) C 2 (X) 
fM <hM 
(«) 
(6) 
In (6) substitute — L x /L 2 for and — M^Mg for ju, when we obtain a quartic 
equation for the locus of Q representing a quartic curve having double 
points at 0 : and 0 2 . 
The biquadratic relation (6) at once indicates the genre of the curve. 
The existence of the double points is deduced analytically in Cor. 1, 
geometrically in Cor. 2 ; and the six possible varieties of these are 
enumerated in Cor. 4. 
§ 23. Prop. XV. 
If P ± Q and P 2 Q coincide simultaneously with 0-fi v the quartic de- 
generates into the straight line 0 1 0 2 and a cubic curve through 0 1 0 2 
devoid of double points. 
Cor. 2. This can happen when l x and l 2 cut on 0 1 0 2 and a -f /3 = ir. 
Cor. 3. Also when a + /3 = ir and l 3 makes an angle a with 0 1 0 2 . 
For, when P 1 comes to lie on OjOg, R goes to infinity on l 3 , while OjQ 
and 0 2 Q coincide simultaneously with 0 1 0 2 . 
Cor. 4 . In particular this will happen when a = /3—^-, and either l x and 
l 2 intersect on C^Og, or l 3 ± r C^Og. 
VOL. xxxvi. 
8 
