REAL GROUPS. 255 
A comparison of the tables of Lie and Meyer show them to 
be practically identical; in fact, Meyer has only put Lie’s in- 
finitesimal notation into finite form. A comparison of the 
present table with Lie’s table shows some results worthy of 
notice. 
The groups numbered (18) and (19) in Lie’s table are the 
groups G,(AA’), and G,(l/’), for all values of r. These are, 
in general, of type I, second class, but they include also the 
groups G,’(AA’) and G,'(ll’) of type II, first class. The ex- 
istence of these latter groups would hardly be suspected from 
Lie’s or Meyer’s formule. 
Again, Lie’s group (7) is in general, G,(A/), for all values 
of r, but it also includes the groups G,/(Al) and G,/(Al’). 
The existence of the latter as distinct groups is unknown to 
Lie’s theory. 
The group G,(Al),_., (8) of Lie’s table, is only a special 
case of G,(Al),, (7) of the same table; and, though its struc- 
ture is somewhat peculiar, it is doubtful if it is worthy of 
special mention in the list. 
$10. Groups of Real Collineations. 
Thus far in treating collineations in a plane we have con- 
sidered the most general case where variables and parameters 
are complex numbers. We shall now examine the special 
ease of real collineations, 7. e., those that transform real 
points into real points. 
309. The Real Group G,. A real collineation is repre- 
sented analytically by the equations 
, _ arr hyte _ a2a+ bry tee 
Tae Bethe OU aa Pe Pos" (1) 
where variables and coefficients are all real numbers. If the 
coefficients are all real, the point (2,, y,) will be real when, 
and only when, (#,y) is areal point. The resultant of any 
