370 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
and the loci of the twofold centre and of the one-with-twofold centre respectively are 
determinate curves, the former a conic, the latter a cubic. The critic centres corre 
sponding to the entire series of satellite lines which pass through a certain fixed 
point lie on a cubic, which when the fixed point lies on the line x + y + z = 0, here 
the line infinity, degenerates into a conic called the “Harmonic Conic,” and when one 
of the critic centres is known the other two are determined as the intersections of 
the harmonic conic by the polar of the given critic centre in regard to the twofold 
centre conic. 
53. For establishing the theory of the groups of the hyperbolas A, it is necessary 
to consider the geometrical forms of the several curves which have been just referred 
to ; viz. this is to be done, on the assumption always that the line x + y + z = 0 is 
the line infinity, for the Redundant Hyperbolas taking the lines x — 0, y = 0, z — 0, to 
be real lines and for the Defective Hyperbolas, taking them to be one real and the 
other two imaginary. The formulae of the memoir above referred to, are in their 
actual form adapted to the former case, but they can of course be transformed so as 
to adapt them to the latter case. I proceed to examine the two cases separately. 
The hyperbolas A Redundant (See Jig. 1). Article Nos. 54 to 71. 
54. Every thing is symmetrical with respect to the three asymptotes, and to fix 
the ideas and without any real loss of generality we may consider the asymptotes as 
forming an equilateral triangle. Taking the perpendicular distance of a vertex from 
the opposite side as unity, the absolute magnitudes of the coordinates may be fixed 
by assuming x + y + z = 1; x, y, z will then denote the perpendicular distances of a 
point from the three sides respectively. If the coordinates of a point are proportional 
to a, ¡3, y, then the absolute magnitudes are of course 
« /3 7 
a + /3 + y’ a + /3 + y’ a + /3 + y'' 
the point may be spoken of indifferently as the point (a, ¡3, y) or the point 
/ a £ 7 \ 
\a. + ¡3 + 7 ’ a + /3 + 7 ’ a + /3 + 7/ ’ 
and it is sometimes convenient to use both of the two notations; thus the Harmonic 
point (that is, the point the harmonic of infinity in regard to the triangle) is the 
point (1, 1, 1) or (i, T). The lines y — z = 0, z — x = 0, x — y = 0, which are the lines 
joining the harmonic point with the three vertices respectively, are in the case of the 
equilateral triangle the perpendiculars from the vertices on the three sides respectively,