94
59
394] ON A LOCUS IN RELATION TO THE TRIANGLE,
ten*
0),
vill
ed.
so that we have identically
- ABCFWH* □ = IPabcfyh-V,
and the conditions V = 0, □ = 0 are consequently equivalent.
The condition
1 _ ^ _ 1 1 - =0
abc af 2 bg n - ch 2 +fgh *
is the condition in order that the function
S’ 1' ?• r V H“’ H
may break up into linear factors; the function in question is
be ca ab Y
which is
( 7 be ca abY \„
r h 7' j- if *)■
( AUG
jyz+ -zx+ yxy) ,
so that the condition is, that the conic
(a, b, c, f g, h\x, y, z) 2 + 2 j^.yz + ^zx+ yxyj = 0,
(which is a certain conic passing through the intersections of the Absolute
ABC
(a, b, c, f, g, Ji$x, y, z'f = 0, and of the locus conic ~^yz-1— zx + y xy = 0) shall be a
J 9 ' l
pair of lines. Writing the equation of the conic in question under the form
(a, b, c,y, j, ^jj#, y, zy = 0,
the inverse coefficients A\ B r , O', F', C, IF of this conic, are
( Abc JLJ\J\AJ \y\AJ\J IWV/ p
~ ^
Abc Bca Cab abc
ST h *
C — 7/|
M" M * M r
so that we have F' : G' : H' = F : G : H. Hence, if in regard to this new conic we
form the reciprocal of the triangle (x = 0, y = 0, z = 0), and join the corresponding
angles of the two triangles, the joining lines meet in a point which is the same
point as is obtained by the like process from the triangle and its reciprocal in regard
to the Absolute. But I do not further pursue this part of the theory.
It is to be noticed that the conic
ABC A
-Z yz + - zx + T xy = 0,
/ 9 h
8—2