130 
Proceedings of the Royal Society of Edinburgh. [Sess. 
SECTION II. 
Wherein Curves are Investigated such as may be Obtained 
from Certain Others by the Use of Given Angles. 
§ 42. Prop. IV. 
With data similar to those in Prop. V of Part I, viz. 0 V 0 2 fixed 
points, angles at P and 0 2 in quadrilateral PR0 2 Q constant. Let P lie 
Fig. 38. 
on a straight line l. But let R now lie on a curve G n of degree n. Then 
Q will generate a curve of degree 3 n. 
Bern. 
Let Q lie on a straight line l v and P on its locus l, then R will generate 
a cubic * cutting C n in 3 n points R p R 2 , . . . R 3n} to which correspond on l t 
the 3 n points Q p Q 2 , . . . Q 3ji . Hence l x cuts the locus in these 3 n points. 
/. etc. Q.E.D. 
Cor. 1. Construct on 0 X 0 2 a circle containing an angle equal to O x PQ 
and cut by l in two points A and B. To each of the n points in which 
OjA cuts C n corresponds the point 0 2 , and similarly for OjB. Hence 0 2 is 
a 2^-ple point on the curve. But O x is not in general a point on the 
new curve. 
Cor. 4. If O x is on C» the degree of the new curve is less by 2, if 0 2 is 
on the curve less by unity. If both points are on C n the new curve is of 
degree Sn — S. 
Cor. 5. If of the three vertices P, Q, R of 0 2 RPQ one is restricted 
to lie on a straight line, a second on a curve C n , the remaining vertex 
generates a curve C 3n of degree 3 n. 
§ 43. Prop. V. 
If R in the preceding is restricted to lie on a curve C ni and P on a 
curve Cm, then Q generates a curve C 3mn of degree Smn. 
* This cubic has a double point at Oi and passes through 0 2 . 
