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158] A SIXTH MEMOIR UPON QUANTICS. 585 
It follows that the two forms 
(a, b, c^x, y) 2 {a, b, c\x, y') 2 cos 2 6 - {(a, b, c\x, y'fycc', y')} 2 = 0, 
(a, b, c\x, yf {a, b, c\x\ y') 2 sin 2 6 — (ac - b 2 ) {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 6; or, if we please, that 6 is the 
radius of the circle. 
212. When 6 = 0, we have 
xy' - x’y = 0, 
an equation which expresses that {x, y) and {x', y'J are one and the same point. When 
6 = \ir, we have 
(a, b, c\x, yjx', y) = 0, 
an equation which expresses that the points (x, y) and (x', 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: 
the definition of a circle is consequently simplified; viz. any point-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 points 
measures the distance between the two points. As regards the analytical expression, in 
the case in question ac — b 2 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 expression for the distance. Suppose that the equation of the Absolute is 
(qx - py) 2 = 0, 
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 (x\ y') the expression 
xy' — x'y 
(qx-py) {qx' -py') 
or, introducing an arbitrary multiplier, 
(q* -pfi) {xy' - cc'y) 
{qx-py) {qx-py')' 
which is equal to 
fix — ay fix' — ay' 
qx — py qx'—py' 
C. II.