GEOMETRY OF AN AXIS OF HOMOLOGY. 79 
Hence as O moves along the Steiner ellipse, the focus 
F traverses the circum-circle and the point D the secant- 
conic, while the sides of the triangle ODF pass through 
the fixed points P, P;P3. 
These points I propose to call the parabolic points of 
the triangle of reference. 
The axis of P, has for equation 
a cos A (b*—c*) +B cos B (c? - a*) + cos C (a*§—b’)= 0 
and is the line joining the orthocentre of the triangle of 
reference and the point (a tan A, } tan B, c¢ tan C). 
The axis of P, has for equation 
a sin A (6? —c?) +3 sin B (c*—a’)+y sin C (a? -0?)=0 
and is the line joining the centroid and the Symmedian 
point of the triangle of reference. 
The axis of P, has for equation 
a cot A (b?—c’) + cot B (c?—a*) +y'cot_C (a? —6*)=0 
and is the line joining the points (tan “A, tan B, tan C) 
and (a* tan A, 6% tan B, c* tan C). 
AS IDAN 
§ 9. The locus of the point whose axis of homology 
is parallel to Z = 0 is the conic 
Ao (bBo —cyo) - Bo (Cy — Aap) xYo (aay — 530) 
MEMES a SIS ese, ee 
p 6 eae SF 
which will afterwards be referred to’as the parallel-conic. 
It passes through the poimt O and the centroid of the 
triangle of reference. 
If the orthocentre of the triangle of reference be the 
point apBoyo, the parallel-conic reduces to Kiepert’s 
hyperbola, : 
sn(B— UC) sin(C— A) ae (A — B) 
£ ws 
§ 10. To find the equation of the pair of tangents to 
S, which are parallel to the line 
