INFINITESIMAL COLLINEATIONS. 291 
ine separately the five different types of plane collineations. 
We shall easily dispose of types V, IV and III, but it will 
be necessary to treat types I and II at greater length. 
343. Type V. There are ~‘ real collineations of type V 
which readily fall into ©* one-parameter groups so that each 
of these finite collineations belongs to one and only one such 
subgroup. Hence we need only to discuss the generation of 
the finite collineations in one subgroup, say RH,/(A1), from 
the infinitesimal collineations of the group. The parameter 
of the group RH,'(A1) is t, which varies from — ~ to +, 
hence the structure of RH,'(A1) is precisely the same as that 
of pG,(A). We may therefore apply the results found above 
for pG,(A) directly to RH,'/(Al). The group of elations 
RH,(Al) contains two infinitesimal elations corresponding 
to the two values of t=+ 45; each of these infinitesimal col- 
lineations generates its corresponding subdivision of the 
group. The general statement may now be made as follows: 
THEOREM 15. Each real collineation of type V may be gener- 
ated from one and only one real infinitesimal collineation. 
344. Type IV. There are ~’ real collineations of type IV 
in the plane which fall into ©’ one-parameter subgroups 
RH,(A',1), so that each perspective collineation of this type 
belongs to one and only one such subgroup. An examination 
of one of these subgroups, say RH,(A,/), shows that it is 
identical in structure with the group hG,(AA’). Hence we 
may formulate the results immediately. 
THEOREM 16. Each real collineation of type LV, for which % is 
positive, may be generated from one and only one real infinitesimal 
collineation; no such collineation, for which # is negative, can be 
generated from a real infinitesimal collineation. 
345. Type Ill. There are ~° real collineations of type 
III in the plane which fall into ~* three-parameter groups of 
the kind RG,’(Al) in such a way that every collineation of 
type III belongs to one and only one such group. Each 
group RG,’ (Al) breaks up into ~* one-parameter subgroups 
