[158 
158] 
A SIXTH MEMOIR UPON QUANTICS. 
583 
208. Write for shortness 
y, zf = 00, 
(a, y, z\x', y, z) = 01 = 10, 
&c., 
then we have identically, 
00, 
01, 
02 
= K 
x , 
y , z 
10, 
11, 
12 
x , 
f, * 
20, 
21, 
22 
// 
X , 
y", z" 
and if the determinant on the right hand vanishes, that is if (x, y, z), (x\ y, z'), ix", y", z") 
are points in a line, then we have 
00, 
01, 
02 
10, 
11, 
12 
20, 
21, 
22 
an equation, which, as already remarked, is equivalent to 
, 01 . 12 , 02 
COS -1 - + COS -1 , ■ = COS -1 -==—7= . 
Voo Vll Vll V22 V00 V22 
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 formulae of some complexity which are introduced out of their 
natural place. 
■ibed 
On the Theory of Distance, Nos. 209 to 229. 
209. I return to the geometry of one dimension. Imagine in the line 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 con 
sidered as thus inscribed in the Absolute is said to be a point-pair 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 determined 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