34 ME. A. CAYLEY’S SIXTH MEMOIE LTOX QUAXTICS. 
the definition of a circle is consequently simplified ; viz. any pomt-pair whatever may 
be considered as a circle having for its centre the harmonic of the Absolute with respect 
to the point-pair ; we may, as before, divide the line into a series of equal infinitesimal 
elements, and the number of elements included between any two pomts measures the 
distance between the two points. As regards the analytical expression, m the case m 
question ac—V vanishes, or the distance is given as the arc to an evanescent sine. 
Reducing the arc to its sine and omitting the evanescent factor, we have a finite expres- 
sion for the distance. Suppose that the equation of the Absolute is 
{qx—py^—^, 
or what is the same thing, let the Absolute (treated as a single point) be the point {p., q\ 
then we find for the distance of the points {x, y) and (.a/, y') the expression 
xy^ — a/y • 
[qx-py] [qx^-py') 
or introducing an arbitrary multiplier, 
{qct-p^)[xy'-a^y) ^ 
{qx—py)[qx' -pi/) 
which is equal to 
fix— ay ^x' — ay' 
qx—py qx'—py' 
It is hardly necessary to remark, that in the present case the notion of the quadrantal 
relation of two points has altogether disappeared, and that the unit of distance is 
arbitrary. 
214. Passing now to geometry of two dimensions, we have here to consider a certam 
conic, which I call the Absolute. Any line whatever determmes with the Absolute (cuts 
it in) two points which are the Absolute in regard to such Ime considered as a space of 
one dimension, or locus in quo of a range of points, and in like manner any pomt what- 
ever determines with the Absolute (has for tangents of the Absolute through the point) 
two lines which are the Absolute in regard to such point considered as a space of one 
dimension, or locus in quo of a pencil of hues. The foregoing theory for geometri of 
one dimension establishes the notion of distance as regards each of these ranges and 
pencils considered apart by itself; in order to bring the different ranges and pencils in 
relation to each other, it is necessary to assume that the quadi-aiit which is the miit of 
distance for these several systems respectively, is one and the same distance lor each 
system (of course, when, as in the analytical theory, we actuaUy represent the quadrant by 
the ordinary symbol the above assumption is tacitly made ; but substituting the thing 
signified for the definition, and looking at the quadiuiit merely as the distance bet«-een 
two points, or as the case may be, lines, harmonically related to the point-pau, or as the 
case may be, line-pair, constituting the Absolute, the assumption is at once seen to be an 
assumption, and it needs to be made explicitly). But the assuniption being made, the fore- 
going theory of distance in geometry of one dimension enables the comparison not only 
