182 THE ROYAL SOCIETY OF CANADA 
.”. à”, b’ and ec’ are the harmonic conjugates of a”, b” and ec” and 
concur in the point of index 
I~ létm letne 
mans 
and .‘. of equation 
which point is the pole of p.; 
The points a’.b”.c”, a”.b’.c” and a”.b”.c’ are the harmonians 
of a’.b’.c’ with respect to the triangle abe. 
IX. To determine p, the numerical measure of the distance between 
P and Q, two points in the plane of ABC, whose Cevan ratios to ABC. 
are given. 
Let a, €2, 6, es and 6; be the indices of A, B, C, P and Q respectively. 
n the index of PQ 
€4= XE, +Vieotzie3 in which x1+y+2=1 
and €; = Xe +Voeo+Z2€3 in which x2+%2+2=1 




then will 
PN = €4€5 = (3122 = Y221) €2€3 == (z1%a = ZoX1) 63€ + (x1Ve — X2Y1)€16 
Also 
__ lee; + Mesa + nee 
an 2A! 
in which 2A’= J/{a*(J—m) (I—n)+... } 
l = SD al M _ ZX: —32X1 Nh __X1Y2—X2V1 
EN i ones ua VE eon NOR 
Em. 2-2 BEN = Kay Ll psi 
DA 0. p ONG ae p 24 NE 
p?=— {a(yo— 91) (S221) +e (go — 21) (ar — 21) H-Go(xe — 1) (2 — 91) } 
On substituting RES for “1, Pee al ER Yi, etC.; pas 
A++ A++ 
obtained in terms of a, b, c and the Cevan ratios of P and Q. 
The following resolution gives the value of p? as the product of 
two determinants. 
Let m, 72, 73 be the indices of BC, CA, AB and 6 be the numeric of 
(PQ Z AB), then will 
A= he =e eee 
