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