GROUPS OF TYPES II, [II. 249 
ko—1 Be! hates eile 19 
ohare elas -, 
bse the existence of the group G,’ (AD) is verified. 
302. The Group G,(A'l’). Let T’ and T,’ be so chosen 
that their invariant figures have the point A’ and the line 1’ 
in common; and let A’ be the origin and /’ the x-axis. The 
normal form of 7’ reduces to 
yrisgigDapet tin nt 
A 
e+ Pied 
oe 1—k t ‘SIPs z 
ee, )s 
ce! c! A : 
(78) 
* y 
Yi = iar t CPST 
k+ e+ (+i \y 
A c! c! A 
A simplification of the normal form results if we set Le =, 
It was proved in chapter II, Art. 136, that, if t and t’ denote 
the parole constants along l’ and through A’ respectively, 
t’= 7,9 t Where 9 is the angle/AA’. In the above normal 
form A is the distance between the inyariant points and c’ 
the sine of the angle between the invariant lines; _ hence t’ in 
the equation t/=“" is the parabolic constant through A’. 
c! 
Making this substitution equations (78) reduce to 
a+ tly y 
1-—k t! c k-1 C= (Same denominator. ) Th 
PG SS I oe 
Writing 7,’ in the same form and eliminating, the resultant is 
also in the same form with the following conditional equa- 
tions: : 
eer iles. (yao ote 
Th |o> Wits ki ki (1—k) 
(2) ek see 
a! Co 2 ka—1 1 1 
atta a at (g-a)+ 
1 C1 — Cue j/k—1 
— t/ tal 3 
Hen + +o ( x ee 1 ( a ) 
Hence the existence of the group G,/(A’1) is verified. 
t= 
k+ 
(79) 
