GEOMETRY OF AN AXIS OF HOMOLOGY. 73 
Let the vertices of the triangle formed by L,L,L, 
beA, B,C). Since the sides of this triangle touch the 
conic Sj, its vertices and those of the triangle of 
reference lie on a conic. 
The coordinates of 4, B, C, being respectively 
(— a, 2B, 270) (2a) — Bo, 270) (2a, 2Bo — Yo) 
it 1s seen by inspection that the points 4, B,C, lie on 
the conic S45. 
In general, if any three points (a,)7,) (a43Boyo) 
(a;3y3) be taken on the line L, the vertices of the 
triangle formed by their axes lie on the conic 
a) 4945 B PoBs Y17V273 
5 sF =) ai ne 
ay2a Bo78 Yo2Y 
§ 3. Let any point a’B’y’ be taken on Sy; its axis 
of homology is 
ead we also have 
° Ao Bo Oe 0 
from which it follows that the axis of a’B’y’ passes 
through ap3o7o. Hence the conic S, is the locus of 
points whose axes of homology pass through apGoyo- 
Hence the line Z and the conics S; and S, may all 
be regarded as having been generated from the point O 
(aoBoyo) 
The axis of homology of the point of intersection of 
the conics 
Oa PO Ly 
a B oy; * 
is the line joining the generating points (a;Pyyi) and 
(as[32y2). 
§ 4. If aBoy. be the Symmedian point of the triangle 
of reference, the in-conic S, becomes the Brocard 
ellipse a is ies 
peta 
