234 THEORY OF COLLINEATIONS. 
‘through A’. This proves the existence of both groups H,(A ) 
and H,(AA’). 
All perspective collineations having a common vertex A 
form a group H,(A); all of these whose axes are concurrent 
form a group H,(AA’); all which have both vertices and 
axes in common form a group H,(A,/). 
281. No Group H,(A'l’). We wish now to use the 
analytic method to disprove the existence of a group which 
might be mistakenly inferred from the geometrical point of 
view. It is plausible to infer that the ~’ perspective collinea- 
tions whose axes are concurrent through A’ and whose vertices 
are on /’, collinear with A’, form a group, Fig. 28(c). To 
test this we have only to write down two such collineations 
and form their resultant. The group property is proved or 
disproved according as the resultant is a perspective or a 
non-perspective collineation. 
Taking A’ for the origin and /’ for the z-axis, S reduces to 
the form 
x p(i-k) «+ky 
Sy a = 15 SES y= 1k 
1 ee Le 
Sf= SS = 7= = 
+ Pp B v B Yy I B B Yy 
(71) 
where p = - = tan 9, the slope of the axis of S. 
A second collineation S, of the same set is 
uy ; pi (1-k1) Hm +hy. 
S:a= isi Sy y= tk 1—k . i 
a (Aa eect PCN a Ge A, Se os Bae (71’) 
1 1 yy 
The resultant is 
=e Ah 2p _k 
ae |p 1—k, pcs k) (t=h)] 
ky (Se ey ees (ile) 
B, B, \ i y 
a) 
=r ) 
Pi B, 
B 
\ 
\Pa (1—ky) + oki (1=k) tb otkhy 
Ys = 1=k  1-=k _ (=k) (t=) ) ti [B(GT } 
SED a ple t— SS 
\ 
B B, B, Wasa aie ee 
