1915-16.] The “ GJ-eometria Organica ” of Colin Maclaurin. 97 
magnitude /3. If 0 X P and 0 2 R intersect in R on a line V, then the point 
Q traces out a cubic having a double point at 0 2 . 
Let 0 1 be chosen as origin, and let RP have the equation 
V = tx (1) 
Then PQ has an equation of the form 
L 0 + ^L 1 + ^ 2 L 2 = 0 ..... (2) 
while 0 2 R and therefore 0. 2 Q has an equation of the form 
M 0 + = 0 (3) 
The elimination of t from (2) and (3) leads to a cubic with a double 
point at 0 2 . Also 0 2 Q and O x P cut in a conic. Cf § 18. 
A geometrical construction for the tangents at the double point is also 
given. 
[It is at once obvious that Maclaurin’s generation of a singular cubic is 
the simplest case of the standard generation of these curves, which may be 
stated geometrically as follows : — 
Fig. 7. 
In the quadrilateral RPQ0 2 the angles at P and 0 2 are constant, and 
and 0 2 are fixed points. Let R be on a conic that passes through O x 
and 0 2 , or on a straight line. Let PQ be constantly tangent to a conic 
whose focus is at O r Then P lies upon a circle (or a straight line, if the 
conic is a parabola). See the Theory of Pedals in Part II. 
VOL. xxxvi. 
7 
