384 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
The corresponding positions of the critic centres on the hyperbola are 
On one branch of hyperbola. 
A lt A 3 , each at infinity, 
A, B s , 
C 13 , a twofold centre, 
On the other branch. 
A 2 , the harmonic point, 
B,, 
Co, a one-with-twofold centre, 
Mo, a real centre, 
{ O's, a one-with-twofold centre and 
C\ 2 , a twofold centre, 
{ A and 
A (Asymptote-Point), A, 
E 3 and E 1 , E.,, 
[A 3 , at infinity, and 
lA^, at infinity, A 3 , harmonic point. 
93. For inclination =30 the positions are (AC)MC'DE(A'C), viz. 
AC, at infinity, touches envelope at infinity, 
M, between (AC) and C', 
C, touching upper branch of envelope, 
A passing through asymptote point A, 
E, passing through crunode of envelope, 
A'C, at infinity touches envelope at infinity. 
The corresponding positions of the critic centres on the parabola are 
A,, (harmonic point) a one-with-twofold centre; A u = A' u at infinity, a twofold 
centre, 
Mo, real centre, the other two centres being imaginary, 
C\o, a twofold centre, C' 3i a one-with-twofold centre, 
A, (asymptote point), D 3 ; D 2 , 
A, E 3 ; E 2 . 
A i3, = A 13 , at infinity, a one-with-twofold centre; A 2 (harmonic point) a twofold 
centre. 
94 For inclination =tan~ 1 2, the positions are AMC'D (EC")NA', viz. 
A, at infinity, 
M, between A and C', 
C', touching upper branch of envelope, 
D, through asymptote point A, 
EC\, touching upper branch of envelope at crunode N between EC" and A', 
A', at infinity.