1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 121 
§ 33. Prop. XXIII. 
Let the intersection of P 1 0 2 and P r _ x P r lie on the straight line 
lx + my ■+■ n = 0 . . . . ( 1 ) 
and Q will generate a curve of degree n + r. 
Let be given by the equation 
M 0 + /xMj = 0 
or 
xA 1 (y.) + yB 1 (y) + G 1 (/I) = 0 ( 2 ) 
Then P r _iP r has an equation of the form 
X fr(t) + y<t>r{t) + = 0 
The condition that (1), (2), (3) be concurrent is 
V m n 
a i(/a) 
f\t) <f>r(t) 
wo 
= 0 . 
(3) 
(4) 
Hence fi = a rational function of t of degree r in numerator and denom- 
inator, and the equation to 0 2 R 2 may be written in the form 
N 0 4- ^N 1 + . . . + f 'N r = 0 (5) 
But P n _iQ has an equation of the form 
M 0 + #M 1 + . . . +t n M„ = 0 (6) 
The equations (5) and (6) therefore give for the locus of Q a unicursal curve 
of degree n+-r. 
Cor. 1. The line (6) envelops a curve of class n. Hence n lines of the 
system pass through 0 2 , so that 0 2 is an ?i-ple point on the locus of Q. 
Cor. 3. When of the points RiQ?! . . . P n-1 all but one lie on fixed 
straight lines, the remaining point generates a curve of degree n + r. 
Cor. 5. By variation of n and r subject to the condition n J r r= constant, 
we may deduce a variety of ways of drawing curves of degree n + r. 
Cor. 6 is not correct. 
Maclaurin states the following generation of a quartic: — 
Fig. SB. 
