388 
ON THE CLASSIFICATION OF CUBIC CURVES. [350 
The satellite line is not parallel to the asymptote, and the different positions give 
the following six of Pliicker’s Groups, viz. 
XVIII. Satellite line cuts upper, cuts lower, branch of envelope. 
XX. „ „ „ „ and passes through 
asymptote point. 
XIX. Satellite line touches upper, cuts lower, branch. 
XXI. „ does not cut upper, cuts lower, branch. 
XXIII. „ „ „ , touches lower branch. 
XXII. „ „ „ , does not cut lower branch. 
Hyperbolas A Defective, asymptote osculating. 
The satellite line is parallel to the asymptote, and we have the six groups, 
XXVIII. Satellite line cuts upper branch, cuts lower branch of envelope, viz. it 
lies above the asymptote point. 
XXIX. Do, Do, but it passes through the asymptote point. 
XXX. Do, Do, but it lies below the asymptote point. 
XXXI. Satellite line touches upper branch, cuts lower branch. 
XXXII. „ does not cut upper branch, cuts lower branch. 
XXXIII. „ „ „ , does not cut lower branch, viz. it 
lies below the asymptote. 
And finally, 
Hyperbolas A Defective, three osculating asymptotes. 
Satellite line at infinity, giving the single group 
XXXV. 
But the division gives rise to a remark such as is made ante No. 71. 
As to the Groups of the Hyperbolas ©. Article Nos. 102 to 104. 
102. Taking 2= 0 as the equation of the line infinity, and x = 0, y=0 as the 
equation of any two lines through the point of intersection of the asymptotes, or 
* asymptote point,’ then the equation of the cubic may be taken to be 
(a, h, c, d\x, y) 3 + kz 1 (Xx + py + vz) = 0. 
103. To determine the critic centres we have 
(a, h, c$x, yY + kz 2 X = 0, 
(b, c, dfpx, yY + kz*p = 0 
%z (Xx + py + vz) + z*v = 0,