268 THEORY OF COLLINEATIONS. 
If k = 7 and A’ +0 in equations (4) these reduce to 
x y 
wy, = B! r= B! ’ (4c) 
1+ ty — — ta aL iy = SZ the 
Al Al 
which are the equations of the group H,'(A), 
B’ 
If je AU=S0e Bap Nie and lim. 
equations (4) these reduce to 
w@ 
=! in 
A’ 
y¥+nt'ax 
 {4+ty+(v—nt)a’ Cs aa ER nig LAA A (4d) 
These last equations are the equations of the group G,’’(Al’) 
although not expressed in terms of the natural parameters as 
in Art. 268. The same method is applicable to the other 
groups of this class. 
318. Second Class. The group G,/(AA/’l’)a, where a 4 0 
or ©, contains only collineations of type II. The structure 
of the other four groups of this class is shown as follows: 
Gi( AAG o* GAA )\ a2 HY Al)- asa. 
GU Naw =o GCA a= Ei CA Saee 
G,(Al)a =@'G/ (AAV )a-— Ay’) Sn) 
CA GCUINGR ea CHCA a ITE (AO) ASSy, Te, 
In this class of groups we meet, for the first time, with so- 
called singular transformations 8. T.; these will be discussed 
later. 
Synthetic Method. In order to show that the group of ela- 
tions H,’(/l’) is contained in G,’( Al’), we choose the collinea- 
tions T’(AA'l’), and T,/(AA,'l’), for which the parameters 
k and t are (at,t) and (a-‘,—t) respectively, and form the 
resultant. The resultant of the two one-dimensional para- 
bolic transformation along l’ is identical, 7. e., t, =t—t=0; 
that through A is parabolic, since the two pencils through A 
have the invariant line / in common and k, = at = z . Hence 
the resultant of T’ and T,/ is an elation S’(A,l’), where A, is 
some point onl’. By varying the values of t and the positions 
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