82 
ME. A. CAYLEY’S SIXTH MEMOIR UPOX QEAXTICS. 
an equation, which, as already remarked, is equivalent to 
01 . 12 _i 02 
The foregoing investigations in relation to the inscribed conic are given for the sake of 
the application thereof to the theory of distance, and it has been necessary to make use 
of analytical formulee of some complexity which are introduced out of their natural 
place. 
On the Theory of Distance, Nos. 209 to 229. 
209. I return to the geometry of one dimension. Imagine in the hne or locus in quo 
of the range of points, a point-pair, which I term the Absolute. Any point-pair whatever 
may be considered as inscribed in the Absolute, the centre and axis of inscription being 
the sibiconjugate points of the involution formed by the points of the given point-pair 
and the points of the Absolute ; the centre and axis of inscription qua, sibiconjugate 
points are harmonics with respect to the Absolute. A point-pair considered as thus 
inscribed in the Absolute is said to be dijyoint^air circle, or simply a circle-, the centre 
of inscription and the axis of inscription are termed the centre and the axis. Either of 
the two sibiconjugate points may be considered as the centre, but the selection when 
made must be adhered to. It is proper to notice that, given the centre and one point of 
the circle, the other point of the circle is determmed in a unique manner. In fact the 
axis is the harmonic of the centre in respect to the Absolute, and then the other point 
is the harmonic of the given point in respect to the centre and axis. 
210. As a definition, we say that the two points of a circle are equidistant from the 
centre. Now imagine two points P, P' ; and take the point P" such that P, P'^ are a 
circle having P^ for its centre ; take in like manner the point P^ such that P , P are a 
circle having P^^ for its centre ; and so on : and again in the opposite dnection, a point 
P' such that P', P' are a circle having P for its centre ; a point P" such that P, P" are 
a circle having P' for its centre, and so on. We have a series of points ...P", P\ P. 
P^, P’*, ... at equal intervals of distance ; and if we take the points P , P mdefinitel) 
near to each other, then the entire line will be divided into a series of equal hrfini- 
tesimal elements; the number of these elements included between any two points 
measures the distance of the two points. It is clear that, according to the definition, 
if P, P', P" be any three points taken in order, then 
Dist. (P, F)+ Dist. (P', P")= Dist. (P, P"), 
which agrees with the ordinary notion of distance. 
211. To show how the foregoing definition leads to an analytical expression lor the 
distance of two points in terms of their coordinates, take 
{a, 1), cXx, yf=0 
for the equation of the Absolute. The equation of a circle ha^fing the point (af, j/) for 
its centre is 
{a, h, c\x, y)\a, h, cjxf, ff cos^ { {a, b, cjx, yjx', y')y=0 ; 
