GROUP STRUCTURES, 265 
H,(AA') = @'H,(A,1) +H/(Al’), 
H,(t) =H; (A,l)-- ay (1), 
H,(A) = 0!H,(A,T)+ H(A). 
The correctness of these structural formule may be proved 
in detail ; we shall give two methods of proof, applying both 
methods to the same case. 
(1) Synthetic Method. Take for example the group 
H,(ll’); we will show that this group must contain collinea- 
tions of type V. Take from the group H, (Il’) two collinea- 
tions of type IV, S(A,1) and S(A,,/) (where A and A, lie on 
/, and/ and /’ intersect in A’), for which the cross-ratios along 
l are kand 1/k respectively. Along / we have two one-dimen- 
sional loxodromic transformations having one and only one 
invariant point A’ in common, hence, Chapter I, Article 32, 
their resultant is a parabolic transformation, leaving A’ in- 
variant, whose parabolic constant t has the value 
=i Naa (1) 
Since both transformations along / are identical, their result- 
ant is also identical. Hence, the resultant of S and S, is an 
elation S(A’l), which belongs to the group H,/( Al). From 
the above value of t we see that by varying the value of k, or 
the position of the points A and A,, all elations in the group 
H,/(Al) are obtained; thus, the group H,(lA‘l) contains 
H,( Al) asa subgroup. In a similar manner the structural 
formule of the other perspective groups may be verified. 
(2) Analytic Method. If A’ be taken as the origin and / 
and l’ as axes of x and y respectively, the normal forms of 
S(Al), S,(A,,/) and their resultant are respectively 
x ky 
S 5 iS : Yy SS 
Cane es ’ 1 a ’ ¢ 
fae sel 14 holy (2) 
B B 
v) ; ao ky Yi 
S;: = ki-1,? Y2 = by ee eee 
