ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
83 
and consequently if (^, y), {x", y") are the two points of the circle, then 
{a, b, cXx, yX^, y') {a, b, c\x', y") 
s/ [a, b,cXx, yf\s/ {a, b, c\a^, y’f\ s / («> b, y'f\s/ {a, b, cjx", y'^f 
an equation which expresses that the points (ir", y”) and {x, y) are equidistant from the 
point (a^', y). It is clear that the distance of the points {x, y) and [x\ y') must be a 
fimction of 
[a, b, cjx, yja^, y') 
>/ (a, b, cX^, yf\/ a, b, cX^', y'f 
and the form of the function is determined from the before-mentioned property, viz. 
if P, P', P" be any three points taken in order, then 
Dist. (P, P')+ Dist. (P', P")= Dist. (P, P"). 
This leads to the conclusion that the distance of the points {x, y), {od , y’) is equal to a 
multiple of the arc having for its cosine the last-mentioned expression (see ante, No. 168) ; 
and we may in general assume that the distance is equal to the arc in question, viz. 
that the distance is 
cos-' (g, b, cXx, yX^'> y') 
n/ [a, b, cXx, yfs/ {a, b, cja/, y'Y 
or what is the same thing. 
sm' 
[ac—b'^) {xy^ — x^y) 
s/ {a, b, cXx, yf\/ {a, b, cXoo', y'f 
It follows that the two forms 
{a, b, cXx, yf{a, b, y'f &— {{a, b, cXx, y)}"=0, 
{a, b, cXx, yf{a, b, y'fsm^d—{ac—b^){xy' — x'y) =0, 
of the equation of a circle, each of them express that the distances of the two points 
from the centre are respectively equal to the arc d ; or, if we please, that ^ is the radius 
of the circle. 
212. When ^=0, we have 
xy' —x!y=0, 
an equation which expresses that [x, y) and [x', X) are one and the same point. When 
we have 
{a, b, cXx, yX^’^ y)=0, 
an equation which expresses that the points [x, y) and (^xf, y') are harmonics with respect 
to the Absolute. The distance between any two points harmonics with respect to the 
Absolute is consequently a quadrant, and such points may be said to be quadrantal to 
.each other. The quadrant is the unit of distance. 
213. The foregoing is the general case, but it is necessary to consider the particular 
case where the Absolute is a pair of coincident points. The harmonic of any point 
whatever in respect to the Absolute is here a point coincident with the Absolute itself: 
