1915-16.] The “ Geometria Organica” of Colin Maclaurin. 103 
Make 
L ROjS = a' 
^R0 2 S = £' 
and let R move on the conic. Then S generates a straight line. 
Let O x S cut PQ in P'. Since PO^P' is constant, and P lies on a straight 
line, the locus of P' is, by the lemma, likewise a straight line; and the 
angle S0 2 Q is constant. Hence we obtain a reduction to Prop. V as for 
the quadrilateral P / S0 2 Q ; and thence to Prop. VII. We may therefore 
assume that R lies on a straight line, and P on a straight line or circle. It 
remains to prove that the locus of P may without loss of generality be 
taken to be a straight line in general, so that we obtain a reduction to 
Maclaurin’s generation of the singular cubic. 
Let the cubic be given by the intersection of 
a 0 x + b 0 y + c Q + t(a x x + b Y y 4 <q) + t\a 2 x + b 2 y + c 2 ) = 0 
or 
Lq + £Lj + tf 2 L 2 = 0 
and 
m Q x + n 0 y +y 0 + t{m x x + n x y +p x ) = 0 
or 
M 0 + ^Mj = 0 
These may be replaced by 
Lq + t\j x + £ 2 L 2 + (A t + B)(M 0 + = 0 . 
and 
Mq + — 0 
( 1 ) 
( 2 ) 
(3) 
(4 
in which A and B are arbitrary. 
The equation (3) will envelop a parabola provided a value of t can be 
found for which (3) is the line at infinity, i.e. so that 
g,q + tct-^ + fia 2 + (A^ + B)(w?q + = 0 .... (5) 
6q + tb -^ + fib 2 + (A£ + B)(tiq + = 0 .... (6) 
Hence t must be such that 
a 0 + ta x + fia 2 _ m 0 + tm x 
6q ■+■ tb x -+■ t^bc, ?Iq + tn x 
( 7 ) 
