FUNDAMENTAL GROUPS. 221 
Me: 
A” and the lines AA’ and A’A” as common tangents, Fig. 
24. When the collineations of type I in the group 
G,(AA’A”) pass over into collineations of type III as their 
limiting forms as A’ approaches A, the pencil of conies 
having double contact at A and A” becomes the pencil S of 
conics having contact of the third order at A. 
The constant a designates (Chapter II, Art. 139) the limit 
of < as the collineation of type I approaches type III. From 
the equations 
[= ((A’—A)?+ (B’—B) )' , 
Fic. 25. 
A BAT 
and Pees el Ove [IE es 
aa |.A” Bi | 0 | Al—2Al + AB! 2R'-B)|? 
we have 
2 aa ee ay he 61 
i LSE (61) 
| Av —2A!'+ A B’—2B'/+B 
