1915 - 16 .] The “ G-eometria Organica ” of Colin Maclaurin. 95 
so that O'Q has the equation 
y(£ -a-7] tan f3) = {x- a)(rj + £- a tan (3) 
or 
- x — a tan (3) - y(y tan f3 + x - a) - a(y - x - a tan 13) = 0 . 
Also OQ has the equation 
i(y - x tan a) - rj(y tan a + x) = 0 
So that if P traces out the line 
Q traces out the conic 
A 
y - (x - a) tan (3 
y — x tan a 
A£ + B?7 + C = 0 . 
-B 
y tan [3 + x- ci 
y tan a + x 
C 
- a(y — x — a) tan (3 
0 
= 0 
passing through the fixed points (0, 0), {a, 0) ; and 
( 
a tan (3 
tan a - tan f3 ’ 
a tan a tan /?' 
tan a - tan (3, 
(3) 
( 4 ) 
(5) 
( 6 ) 
Denote the last point by O". 
The three points are the singular points of the transformation, and are 
as in the figure. 
When a = 3, 0" is at infinity. The curve cannot be an ellipse, and is 
a parabola when l is parallel to 00'. When l passes through one of the 
points 0, O', O", the conic reduces to a straight line. 
SECTION II. 
Description of Lines of the Third Order having a Double Point. 
§ 8. Maclaurin s researches on these curves will well stand comparison 
with the modern theory of these curves, which he may fairly be described 
as anticipating. 
