222 THEORY OF COLLINEATIONS. 
(Paw 
— Ag Aty— AyAtx* 
Passing to the limit we have 
ON =o eed eee 
m dxd?y—dydx 
where F is the radius of the limiting circle through three con- 
secutive points. If the three points AA’A” be ona curve, 
then FR is the radius of curvature at the limiting point A. In 
this case the points AA’A” do not lie in the path-curves of 
the group G,’/( AA’A’’),, but are the vertices of the triangle 
formed by two tangents and their chord of contact. Now in 
the limit the radius of the circle circumscribing the triangle 
formed by two consecutive tangents and their chord is one- 
half the radius of curvature.t Hence a is half the radius of 
curvature at A of each conic of the family of path-curves of 
the group G,’’( AIS). 
This conclusion may be verified directly by calculating the 
radius of curvature at the origin of the conic 
a? + 2hay+4aCy—4ay=0. 
We readily find the radius of curvature at the origin to be 2a 
and thus independent of C, the parameter of the family of 
conics. 
THEOREM 46. The constant a, in the group G,”( A/S) is half 
the common radius of curvature of the path-curves of the group at 
this common point of contact, a. 
267. Geometric Meaning of h. To find the geometric 
meaning of h we find the equation of the polars of a point A’ 
on the «-axis with respect to the pencil of conics given by 
v?+2heuy+4aCy?—j4ay=0. (60) 
The coordinates of A’ are («’, 0) and the polar of («’, 0) is 
a’ (ae’ +hy)=4ay. 
*Goursat, Cours de Math. vol. I, p. 490. 
+Salmon, Conic Sections, 6th ed., art. 398a. 
