376 
OJST THE CLASSIFICATION OF CUBIC CURVES. 
[350 
The satellite line is parallel to the osculating asymptote, say to the base of the 
triangle ; and the different positions give the following six of Pliicker’s groups, viz. 
IX. 
Satellite line above the vertex. 
X. 
99 
through the vertex. 
XI. 
99 
below the vertex, but not cutting envelope. 
XII. 
99 
touches envelope. 
XIII. 
99 
cuts envelope. 
XIV. 
lies below the base. 
And finally, 
Hyperbolas A Redundant, three osculating asymptotes. 
The position of the satellite line is here completely determined, giving one of 
Pliicker’s groups, viz. 
XVI. Satellite line at infinity. 
71. It may be remarked that in this enumeration no account is taken of the 
nodes of the envelope: the enumeration was in fact made by Pliicker by considerations 
relating to the critic centres, but without arriving at or making use of the envelope 
at all: if account were taken of the nodes of the envelope several of the foregoing- 
groups would have to be subdivided according to the different positions of the 
satellite line in regard to these nodes: but the effect produced by the passage 
of the satellite line through a node of the envelope is so slight, that I am inclined 
to think that the enumeration may be properly effected in the foregoing manner, 
without any account being taken of these nodes. 
The Hyperbolas A Defective {See fig. 2). Article Nos. 72 to 101. 
72. If in the formulae for the Hyperbolas A Redundant we write 
\{x + yi) for x, 
%{x-yi) „ y, 
X — pi „ X, 
X + pi „ p, 
then the equation of the satellite line is 
\ {X — pi) (x + yi) + \ {X + pi) (x — yi) + vz = 0, 
which is Xx + py + vz = 0 as before; the equation of the line infinity is x + z — 0, and 
the equation of the curve is 
£ (x 2 + y 2 ) z + k {x + z) 2 (Xx + py + vz) = 0.