1915-16.] The “Geometria Organica” of Colin Maclaurin. 123 
Then R 2 (£) and 
IS) - ^(0 
P 2 (0 QaW 
vanish when £ = a. 
Hence on solving (5) and (6) for aj and y we find 
= / 8 (0(* - a )/W0(* - a ) ! _ _ . (8) 
y = il/ 3 (t){t- a)/cl>2(t)(t- a) f 
etc. 
§ 34. Proj9. XXIV. 
Consider two serrate angles 
0 1 P 1 P 2 . . . P»n.P 
a ii fl 
0 2 Q x Q 2 . • . QnQ 
in which P X P 2 . . . P m lies on m fixed lines , and Q ± Q 2 ... Q n on n fixed 
lines. 
If the intersection of P m P and Q n Q also lies on a given straight line , 
the intersection of 0 1 P 1 and 0 2 Q 1 in general generates a curve of degree 
n + m + 2 possessing an (m+l)-ple point at 0 1 and an ( n-\-V)-ple point 
at 0 2 . 
Let 0 1 P 1 have equation 
y — Xx = 0 . • • • • (1) 
Then P m P has an equation of the form 
xA m+1 (X) + yB m+1 (\) + C m+1 (X) = 0 . • ( 2 ) 
If OgQj has an equation of the form 
L 1 + ^L 2 = 0 ..... (3) 
Q n Q has an equation of the form 
xA n+ ff) + v/B n+I (^) + C w+1 (/)Bo • (‘f) 
Let P m P and Q W Q intersect on 
Hence 
ax + by + c = 0 
a b 6 
Xn+, l(h) 
A n +l(t) 
Vn + ,(A) 
Pn+l(0 
C m+1 (A) 
C n+1 if) 
= 0 
( 3 ) 
( 6 ) 
Substitute yfi for X, and — L x /L 2 for t in (6), when the result follows 
at once. 
Cor. 2. If of the points P^ . . . P m Q x Q 2 . . . QJRT (T being the 
intersection of 0 1 P 1 and 0 2 Q 1? and R of P m P and Q n (l all but one lie on 
straight lines, the remaining point generates a curve of degree n + m + 2. 
