120 
Proceedings of the .Royal Society of Edinburgh. [Sess. 
For P„Q has an equation of the form 
L 0 + £Li+ . . . +P +1 L n+1 = 0 . . . . (4) 
and 0 2 Q, which really makes a constant angle with 0^, has an equation 
of the form 
M 0 + £M X = 0 (5) 
The elimination of t between (4) and (5) leads to an (n + 2)-ic having 
an (n-f l)-ple point at 0 2 . 
[We might state the theorem thus. Given Oj and 0 2 fixed points, and 
the serrate angle 
GjPjP 2 • • • P?iQ0 2 , 
in which P x . . . P n lie on fixed straight lines, the locus of Q is an 
(w + 2)-ic with an (?i + l)-ple point at 0 2 . Or, again, the locus of Q is 
simply a pedal of the envelope of P n Q.] 
§ 31. Prop. XXI. 
Given the serrate angle 0 1 P 1 . . . P n _ x Q (n— 1 lines l v l 2 , . . . C-i) and 
the constant angle P 1 0 2 P 2 which is rotated round 0 2 . If the intersection 
R x of 0 1 P 1 and 0 2 P 1 lies on a fixed line l n , then the intersection Q of P n _ x Q 
and 0 2 R 2 generates a curve of degree n + 1. 
For the equation to P n _iQ is of the form 
Lq + t\j-^ + . . . — f l L n = 0 . . . (1) 
In virtue of l n the parameter of 0 2 R 1 and .*. of 0 2 R 2 is in 1 — 1 corre- 
spondence with t, so that the equation to 0 2 R 2 is of the form 
Mo + ftJo ...... (2) 
The elimination of t between (1) and (2) gives rise to an (^ + l)-ic 
with an n-ple point at 0 2 . 
Cor. The curve may, of course, degenerate and be of lower order in 
its component curves. 
Cor. 6. The angle R 1 0 2 R 2 may be a straight angle, so that R 1 0 2 R 2 is a 
straight line rotating round 0 2 . 
Cor. 7. When n = 3, the curve is a quartic with a triple point at 0 2 . 
§ 32. Prop. XXII. 
If all the points but one of the n + 1 points 
Rj.Pi • • • Pn-iQ 
are restricted to lie on straight lines , the remaining point generates a curve 
of degree n-\- 1. 
The proof is exactly on the lines of Prop. XIII. 
