PERSPECTIVE GROUPS. " 225 
$8. Groups of Perspective Collineations. 
270. In§7 we investigated the fundamental groups of 
collineations of types IV and V, viz.: the one-parameter 
groups H,(A,/) and H,'/(Al). In the present $ we wish to 
determine all varieties of groups which contain only perspec- 
tive collineations. We know from the results of Arts. 123, 
228, that there are ~’ perspective collineations in the plane 
and -that these do not form a group. We also know that the 
system of «” collineations of type IV which have a common 
axis (or common vertex) do form a group. We have also 
learned, Art. 124, that there are ~’ elations in the plane and 
these do not form a group; and a number of similar ques- 
tions have been settled. 
But in the present § we wish especially to show how the 
normal form of the collineation S and S’ may be used to de- 
velop a complete theory of perspective collineations. Weshall 
also make free use of geometric methods. Perspective col- 
lineations are a special kind, and for this reason they are 
specially fitted to illustrate the various methods that may be 
employed. We shall therefore ignore for the most part the 
results already obtained for types IV, V and proceed to inves- 
tigate these types de novo; in so doing we shall sacrifice 
brevity for the sake of ample illustration. The results ob- 
tained will serve as a check upon the methods of $§ 4 and 5. 
A. GROUPS OF TYPE V. 
271. Two-parameter Group H,'(1). Let us take two 
elations, designated by S’( Al) and S,/(A,/) having the same 
axis | and their vertices A and A, on /, Fig. 27(a), and deter- 
mine the character of their resultant. Since S’ and S,’ both 
leave invariant every point on /, their resultant also leaves 
invariant every point on / and is therefore a perspective col- 
lineation. Let t and t, be the characteristic constants of S’ 
and S,’ respectively, and let us consider the effect of S’ and 
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