The collected mathematical papers of Arthur Cayley.

394]

ON A LOCUS IN RELATION TO THE TRIANGLE.

59

so that we have identically
- ABCF2G2H2 o = K2abcf2g2h2V,
and the conditions V = 0, = 0 are consequently equivalent.
The condition
1 1 1 1 2
abc af 2 bg2ch2 fgh
is the condition in order that the function
' b" a' f' c g ax, by, cz 2,
may break up into linear factors; the function in question is
b, be ca ab )2
a, b, c, f,, -j- xy, z\,
which is
A B C
=(a, b, c, f, g, h7x, y, z)2+ 2 ( - yz + zx - x
so that the condition is, that the conic
(a, b, c, f g, h7(x, y, z)2+ 2 A yz+-zx+ Exy =0,
(which is a certain conic passing through the intersections of the Absolute
A B C
(a, b, c,, g, hix y, yz)2= 0, and of the locus conic A yz + - z -s shall be a
pair of lines. Writing the equation of the conic in question under the form
c y, be ca ab 2= 0,
the inverse coefficients A', B', C', F', G', H' of this conic, are
( Abec Bca _ ab abe ab ab )
f ' g h2'fgh gh f gh J
so that we have ': G': H' = F: G:. Hence, if in regard to this new conic we
form the reciprocal of the triangle (x= 0, y=O, 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
A B C
f yz+- zx+ + xy=O,