236 THEORY OF COLLINEATIONS. 
to the first. Two such groups are said to be dualistic to one 
another ; if the invariant figure of the group is a self-dualistic 
figure, the group is called a self-dualistic group. 
As examples, we may cite the following pairs of dualistic 
groups: H,(A)) and 4,(1), H,(’) and Hj(A A’), AAD 
and H,'(l); the one-parameter groups H,(A,/) and H,’(A1) 
are self-dualistic groups. 
This principle holds for all collineations of whatever kind, 
and hereafter, when the existence of a certain group has 
been proved, it will be assumed, without further proof, that 
the dualistic group also exists and has properties dualistic to 
the first. 
284. Resume of Perspective Collineations. In the follow- 
ing list, the structure of all groups of perspective collineations 
is indicated. A dash above a letter indicates that the line or 
point thus marked takes on different positions in the invari- 
ant figure of the group. Thus H,’(l) = ~ H,’/(A1) indicates 
that the point A takes all positions on the line/. Dualistic 
groups are bracketed together, and self-dualistic groups are 
bracketed alone. 
| H,’(Al)f, 
} gl (A, l) ; ’ 
Wee (Le) eee (A) 
|H,/(A) = 'H,'(Al), 
\ H,(Il’) of H,(A,1)+H,'(Bl), 
| H,(AA’) = 01H,(A,1)+H,/(AV), 
\ 7, (7) = »?H,(A,l)+H,'(1), 
| H,(A) o* H,(A,l)+H,'(A). 
| 
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