1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 127 
Let 0 1 0 2 R = i 8 and let 0 2 R cut the given conic in R and R'. Then,, 
when P comes to R or to R', Q comes to 0 1? which is thus a double point. 
Similarly it passes once through 0 2 . 
Cor. 6. If O x Q and 0 2 Q coincide simultaneously with 0 1 0 2 , the curve 
reduces to a conic. 
Cor. 8. Particular cases of Newton’s organic description as a Cremona 
transformation when a = /3= 
Choose Oj as origin, and 0 2 as (a, 0). 
Then, if P is (£ rj) and Q (£', */), 
( 1 ) 
7)' = a) I v ( 2 > 
Thus: 
(L) To 
lx + my + w = 0 
corresponds 
l(a - x)y — mx{a — x) + ny = 0, 
a conic through 
(0, 0) ; (a, 0) ; (a, co ). 
(II.) To the parabola 
y 1 — mx = 0 
corresponds 
x 2 (x — a) 2 + m(x — a)y 2 = 0, 
or 
my 2 = x 2 (a — x). 
(III.) To the rectangular hyperbola 
corresponds 
x 2 — y 2 = mx 
y 2 (a — m — x)=x 2 (a — x). 
(IV.) Oj at infinity on the cc-axis ; 0 2 , as before, (a, 0) ; P0 2 Q = 
O x PQ collinear points ; xy =p the locus of P. 
Let P be the point (p/rj, rj), .'. 0 2 P has equation 
vh = v(x - a)/(p - ay) (1) 
The equation to 0 2 Q is 
( 2 ) 
v 
