[cLasHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 179 
ss Peres + dese + Tree 
(p-+q—")A"B" — pA" Q—gPB" 
Peres + Qesei + Tee) 
pQB"+qA"P—rA"B" 
and PAB=PAQ, 
ne Peres + ése1 + re1e2 
2A 
lesez + Mése + nee 
N'{a(1—m) (—n)+... } 
= alm+bmm+cnns 
a J{a2(l— m) (J—n)+... } 
V. The condition that a point shall lie on a line is that the product 
of their indices shall vanish, therefore using the notation of proposi- 
tions I and III, if D lie ond 
LE (Ne Huet vez) (pees gee: + Tee) 
2(A+u+v)A 
We 2Sye (a 2,200 
2(A+u+?) 
prt+qutry=0 
D +mu+nr=0 
This is the point equation of the line of index 7 and the line 
equation of the point of index e, in the former case the ratios 1:m:n 
are constants, in the latter case the ratios \ :u :v are constants. 
VI. The point P and the line p are said to be relatively pole and 
polar with respect to the triangle ABC, if the Menelaän points of 
abc;p are the harmonic conjugates with respect to A, Band C of the 
Cevan points of ABC; P, i.e., if 
jade Arex + He + 16 
No ea A 
A tees + 1 tee, lee 
V{æ(1 = i) Ay “te y, t+ aiisike } 
ARNO 
NT 
M1 V1 
then will indp = 

will be the point equation of p 
