SINGULAR TRANSFORMATIONS. AUS 
Now tand t, may be so chosen that t+ t, # 0 while a‘**: = 7 
in an infinite number of ways, one for each integral value of 
n in equations (8). Hence, there are ~’ such collineations 7” 
in G,’(Al’). But since » can have only integral values, these 
collineations of type III do not form a continuous system. 
323. The Discontinuous Group dG’ (Al’). We shall now 
examine this discontinuous system of collineations and see if 
it has one or both of the defining group properties. Let 
T’(Al’) and T,’(Al’) be two collineations of this system 
whose parabolic constants along /’ are t and t,. Their result- 
ant is of type III, since both belong to the group G,’(Al’). 
Since ¢ and t, are both of the form 
2nx (A+ilogr) 
6? + log? r (10) 
t=p+iq= 
where 7 has only integral values, we see that t., the sum of 
tandt,, isof the same form and n,="-+n,. Hence 7,//(Al’), 
the resultant of T”’ and T,’, belongs also to the discontinuous 
system, and this system of singular transformations in 
G,(Al’), has the first group property. 
Let T” be one of the collineations of the discontinuous sys- 
2n7(¢+ilogr) The 
#2 + log? r 
parabolic constant of 7’’~', the inverse of 7T”’, is —t and hence 
is of the same form as ¢t with — n for n in (10). Hence, the 
inverse of every collineation in the system is also in the sys- 
tem; thus this system of singular transformations has the 
second group property. 
Since this discrete system of singular transformations in 
G,'(Al’), has both group. properties, it is a group, but a dis- 
continuous subgroup of G,’(Al’). We shall designate it by 
iG) (CAL). 
THEOREM 1. The system of singular transformations of type 
IILin G@s/(A/’) forms a discontinuous group, d G”(A/’). 
tem whose parabolic constant along / is t = 
324. Singular Transformations in G,/(ll'),. If T/(AA') 
and T,'(A A,’/l’) be chosen from the same group G.’(//’), and sueh 
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