[cLasHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 191 
XVII. If the triangles abc, efg are in plane perspective, ae, bf 
and eq being their axial points, the lines af | be, bg | cf and ce | ag 
form a triangle in concentric perspective with abe and efg. 
Let D be the centre of perspective of abe and efg, also let H=af, 
I=bg, K=ce, L=ag, M=be, N=cf, qa=HM, o=IN, p=KL and 
De Aer + Le + veg 
p+u+v 
AE pe + Le + ves 
p+u+v 
her + Xe + ve 
one ane 
het pete, 
A+ut+y 
Be ind H = xe+(1—x)é 
then will {xe+(1 — x)es } (he: uertve;) (Ne: ue He) =0 
Eliminating x: 
ing k= 
ind G = 
(A — D) e+ rv— op)es 
(À — D)u+Ar—Y 
The substitution (ee;) (uv) (xŸ) does not affect ind à but transforms 
ind f into ind g, therefore it transforms indH into ind L 
ind i 
(Au—ox)eet (A —@)ves 
(A —D)r+Au— x 
The substitution (ee2¢;)(Auv) (ox) transforms ind H and ind L 
into indI and indM, and repeated, into ind K and ind N 
à ind L 
Ta ee et 0 
(u—x)r+Au— ox 
ede a eat ee OS 
(u—x)A+py—xy 
(v—p)\+yy— xy 
Preps re ee Vie 
(v—p)ut+rv— oy 
Let O=AD . HM, then will 
ind K = 
xX (€o-+- Ves) 
ind O=(1—x)e + 
c uty 
Sec. III. Sig. 5. 
