350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
373 
63. Imagine now the satellite line moving parallel to itself through the series 
of positions ABGMDEA'; to simplify the figure these are not delineated in their 
proper positions (but they are merely indicated according to their order of succession), 
and it is to be understood that they have the following positions, viz. 
A, at infinity( x ), 
B, through the vertex B, 
G, touching the envelope, 
M, through the vertex M 3 , 
D, through the vertex A, 
E, through node X of the envelope, 
A', at infinity^); 
then the corresponding positions of the critic centres are 
On one branch of the hyperbola, On the other branch, 
A 3 , at infinity, J.j, at infinity: A 2) the harmonic point, 
B lf B. 2 , 
G 12 , a twofold centre, 
M 1} M 2 are imaginary, 
C 12 , a twofold centre, 
A, A, 
A, A, 
A/, at infinity; A 2 , the harmonic point. 
G 3 , a one-with-twofold centre, 
M 3 , 
C 3 ', a one-with-twofold centre, 
A 3 , at infinity. 
64. For the further explanation of the figure it is to be observed that B 2 , B s 
lie on the line joining the midpoints of two sides; and in like manner A, A on 
the line joining the midpoints of two sides; (the imaginary points M 1} M, are in 
like manner on the line joining the midpoints of two sides): these relations depend 
on the theorem, No. 81, of the Memoir on Involution, viz. that for the satellite lines 
which pass through a vertex (1, 0, 0) of the triangle, one of the critic centres is the 
vertex (1, 0, 0), and the other two critic centres are points on the line — x + y + z = Q, 
or, what is the same thing, x = \. 
65. Again, the point A is 011 the line (a?=l) through the vertex A parallel to 
the base, and the points A, A are on the hyperbola (indicated by a dotted line in 
the figure) (y -1- \){z + £) =-£§; this depends on the theorem Nos. 73 and 74 of the 
1 Strictly speaking a line at infinity is the line infinity, and as such has no definite direction; but we 
may of course consider a line which moves parallel to itself in opposite senses as having for its limit the 
line infinity.