ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
345 
349] 
65. The line \x + gy + vz = 0 may be expressed in the form, 
X V z 
«3 + ^ + y + h(x + y + z)=0, 
(where, ut supra, a + /3 + 7 = 0). The corresponding values of f g, h are 
f : g : h = a 3 (fi 3 — 7 3 ) : /3 :! (7 3 — a 3 ) : 7 3 (a 3 — /3 3 ), 
or, what is the same thing, 
f : g : h = a?(/3 -7) : /3 3 (7 -a) : 7 3 (a -/3), 
or, again, 
/: 9 : /¿ = a 2 (/3 2 -7 2 ) : /3 2 (7 2 -a 2 ) : 7 2 (a 2 -/3 2 ). 
The equation /ir + gy + hz — 0 may be written 
y> 
z 
= 0, 
1, 1, 1 
I 1 I 
a 2 ’ p-' 7 2 
that is, the line in question is 
the point 
the inverse of the point (a 2 , /3 2 , 
the line 
with the envelope. 
the line joining the harmonic point (1, 1, 1) with 
(- - -) 
W ’ /3=’ y'-J ’ 
7 2 ), which is (ante, No. 27) the point of contact of 
X y z 
j 3—; — 0 
a 3 p 3 7 3 
66. The harmonic conic passes through the vertices of the triangle x — 0, y — 0, z = 0, 
through the harmonic point (1, 1, 1), and through the critic centres. Hence if one of 
the critic centres be given, the harmonic conic passes through five given points and 
is thus completely determined. But a critic centre being given, the line joining the 
other two critic centres is the polar of the given centre in regard to the twofold 
centre conic (ante, No. 40), and it is thus completely determined; and the other two 
critic centres are of course the intersections of this line with the harmonic conic. 
Article Nos. 67 to 87. Miscellaneous Investigations. 
67. I demonstrate by means of the last-mentioned formulae a theorem already in 
effect demonstrated by the investigation which led to the three centre conic, viz. that 
the tangents at a node or critic centre, and the lines drawn to the other two critic 
centres, form a harmonic pencil, 
C. V. 
44