1915-16.] The “ Geometria Organica” of Colin Maclaurin. 117 
or 
a x 2 ad 
“G 
ad 
?/ -H d 4- — (ft — a -H — j — 0 
Similarly P 2 R has the equation 
^ + ( ( b --^)- y + U b -L- b + b l) = 0 
On solving (2) and (3) for £ 2 and £ we obtain 
£2 _ aft 2 + /6ft?/ + yft + $y + e 
Aft + B 
£= 
Aft + yu.?/ + V 
Aft + B 
.*. the. equation to the locus of R is 
(Aft + fxy + v) 2 = (Aft + B)(aft 2 + . . . 4- e) 
( 2 ) 
(3) 
( 4 ) 
Cor. 7. If, as before, a = f3 = ^ , and ^ and l 2 coincide, the curve is 
a cubic. 
For, let l x and l 3 cut in D-l and D 3 respectively. 
Then D X E ± r 0 1 0 2 forms part of the locus. 
§ 27. Prop. XIX. 
If in the figure of Prop. XIV Q, P 2 , and R are restricted to lie on 
straight lines, the point P x generates a quartic with a triple point at 0 V 
Take O x as origin. 
Let O x Q have the equation 
and 0 2 Q have the equation 
y — yx = 0 
y = m(ft — a) 
There is .\ a 1 — 1 correspondence between m and /u. 
P 2 R has an equation of the form 
%fM + + Iff) = 0 . 
( 1 ) 
(2) 
(3) 
