FUNDAMENTAL GROUPS. 219 
The resultant, T,’’, belongs to the same set as 7” and T,”, 
and hence the first group property is established for the set. 
The inverse of JT” is gotten by solving the equations of 7” 
for x and y. Thus: 
S 
~+42a(—t), 
f Mi . 
Ins 
(57) 
ll 
ee 
y 
1 
—=—-t> +at?- ht. 
y y 
< 
The equations of 7” and its inverse differ only in the sign of 
the parameter t. Hence both group properties are established 
and the set is a group of three parameters, G,’’( Al). 
263. The Two-parameter Group G,’(AIN). Let T’(aht) 
and T,’’(ah,t,) be two collineations of the group G,’’(Al), 
1.e., let T”’ and T,”’ be so chosen that they are characterized 
by the same constant, a. Then their resultant 7,” is also 
characterized by the same constant, a; for if a,=a in the 
three equations (18), then also a, = a and they reduce to two, 
as follows : 
bart 
Peto. eo) 
Hence in G,’’(Al) if a be kept fixed and h and t be allowed 
to vary, we select from G,’’(Al) ~* collineations which form 
a two-parameter group, G,/’(AlN).* 
On the other hand, if / or t be kept constant in equations 
(18), these three equations do not reduce to a smaller number 
and we have no corresponding subgroups. 
THEOREM 43. The fundamental group Gs” (AZ) contains «+4 
two-parameter subgroups G, (A/V), one for each value of the con- 
stant a. 
264. One-parameter Groups G,//(AlS).* Let a,=a and 
h,=h in the system of equations (58); these equations then 
reduce to a single equation t,=t+t,. Hence if a and h are 
both kept constant and ¢ alone be made to vary, we thus se- 
lect out of G,’’( Al) 1 collineations which form a one-para- 
*The significance of the symbol G, (AlN) will be shown in Art, 266, 
