[cLAsHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 181 
Sy) 
.’. A”, B” and C” are the harmonic conjugates of A’, B’ and C’ and 
lie on the straight line of index 
(N)=tees + (u’) 163; + (v’) lee 
2h 
and .°. of equation 
À 
ones asl 
ee tay 
which line is therefore the polar of P. 
Fhe Ines “A” | BoC A” |B” | Cioand A’-| B’ | Care the 
harmonians of A” | B” | C” with respect to the triangle ABC. 
VIII Given the triangle abe and a coplanar line p to locate the 
pole of p with respect to abe. 
Construct 
b.c |a.p=a”,. c.a | b.p=b” and a.b | c.p=c" 
b.c | b’.c’=a’, c.a|c’.a”=b’ and a.b | a’.b’=c’, 
then will a’, b’ and c’ be concurrent at the pole of p. 
Let «a, ©, €, €”1, €"2, €”3, €1, €2 
and e’; be the indices of b.c, c.a, 
a.b, a.p, b.p, c.p, a.a’, b.b’ and 
c.c’ respectively, then will 
: le; — ne, 
€ 2 ee 
l—n 

and 

eee LO ee ER RE TE ee 
—n m—n —m l—n 
“ pet — 1 — 1 
ind b”.c” = ent au 
—l-°+m-~+n 
Similarly it may be shown that 
bate nee 
é'5 = ——— and €; = 
ae Tm 
