124 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Cor. 4. There is no change in the nature of the curve if, instead of the 
O 7 
intersection of 0 1 P 1 and 0 2 Q l5 we take the intersection of two lines through 
0 X and 0 2 making given angles with 0 1 P 1 and 0 2 Q X (in virtue of the 1 — 1 
correspondence). 
Cor. 5. The number n + m-\- 2 for the degree is a maximum, and may 
not always be attained. 
§ 35. Prop. XXV. 
If the intersection of P 8 _JP S and 0 2 Q ± is restricted to lie on a straight 
line, the point of intersection of P m P and Q n Q is on a curve of degree 
ns-\-s-\-m -\- 1 
For P s _iP s has an equation of the form 
and OgQj of the form 
^A s (X) + 7/B s (X) + C s .(A) = 0 . 
. . . . (1) 
L x + £L 2 = 0 
■ ■ • • (2) 
and 
■ • • • (3) 
But P m P and Q W Q have equations 
-t ybm+i(^-) C m+ i(A) 0 
• (4> 
and 
• ock n+x {t) + yB n+1 (t) + ojm = 0 
or 
2/B«s+s(^) C ns _|_ s (A.) 0 
. . (5) 
and the desired result follows from (4) and (5). 
§ 36. Prop. XXVI 
If the intersection of P m P and Q n Q is on the Hite 
ax + by + c = 0 . 
■ • • • 0> 
then the intersection 
of P r -\I \ and Qs^Qs generates a curve of degree 
tin H- 1) + s(m +1). 
We have the relation 
a b c 
B„ +1 (A) C,„ +1 (X) 
^n+l(0 Bn+l(0 P«+l(0 
= 0 (2) 
while ■P } ._ 1 P r and Q,_iQ s have equations of the form 
X A,/A) ••• j/B/X) I- C,.(/V) - • 0 . 
■ - • (3) 
and 
. • ■ ■ (4) 
XA.(i) + yB,(i> + e t (<) = 0 . 
