178 THEORY OF COLLINEATIONS. 
must be a finite group. (The identical collineation alone con- 
stitutes a finite group.) 
It may be possible, however, to obtain a continuous group 
by imposing simultaneously upon the elements of M two sets 
of quadratic relations R, and FR,’ so related that the two in- 
variant conics have a special relation to each other. The only 
relations between the conics that we need to consider are 
those of contact ; for if two invariant conics cut each other 
at four no three of which are collinear points, these four 
points are invariant and the only collineation leaving them 
invariant is the identical one. We must examine, however, 
all possible cases of contact of two conics. 
All possible cases of contact of two conics have already been 
discussed in Art. 125 and their figures given in Fig. 16. If 
two conics K and K’ have contact of the first order, this re- 
duces by one the number of conditions on the elements of M ; 
but we still have nine relations on eight parameters and no 
continuous group. If K and K’ have second order contact, 
there are eight relations on eight parameters and no continu- 
ous group. But if K and Kk’ have a double contact at two 
points A’ and A”, it is possible that we may have a one- 
parameter group. For among the ©° collineations leaving 
the triangle (AA’A’’) invariant there are ~' that also leave 
K invariant ; these may also leave K’ invariant at the same 
time. That this is the case will be shown in the next §, and 
the group obtained is no other than G,(AA’A’’K). If K and 
K’ have third order contact, there is no continuous group (of 
type I) leaving them simultaneously invariant. It is possible, 
however, to find a system of collineations which interchange 
K and K’; this case will be considered later. 
214. The Limiting Case A(l)=0. In order to make the 
above discussion complete two limiting cases remain to be ex- 
amined, viz.: the group defined by three sets of linear rela- 
tions when the determinant of the three sets of constants 
Art. 207 vanishes, and the group defined by a set of quadratic 
