ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
87 
is assumed that the points of the circle are all of them equidistant from the centre, and 
by this assumption we are enabled to compare distances upon different lines. In fact 
we may, by a construction precisely similar to that of Euclid, Book I, Prop. II., from a 
given point A draw a finite line equal to a given finite line BC, and thence also upon 
a given line through A, determine the finite line AD equal to the given finite line BC. 
Since the unit of distance for pomts on a line is arbitrary, we cannot of course compare 
the distances of points with the distances of lines. The distance of a point from a line 
does, however, admit of comparison with the distance of two points ; we have only to 
assume as a definition that the distance of a point from a line is the distance of the 
point from the point of intersection of the line -with the quadrantal line through the 
point. 
221. As regards the analytical theory, suppose that the point-coordinates of the two 
points of the Absolute are (^, r), ^o): then the line-equation of the Absolute is 
so that we have C=2rro, f=qr^-\-rq^^ (S=?Po+F’o, 
and thence K=0 ; but 
where ob\iously 
K(a, h, c,f, q, hjx, y, zf 
X , 
y-> 
Jpo, 
^0? 
X , 
q^i 
is the equation of the Absolute line. 
222. The expression for the distance of the two points {x, y, z), (P, y\ z') is given as 
the arc to an evanescent sine ; but reducing the arc to its sine, and omitting the 
evanescent factor, the resulting expression is 
s/2 
X, y, z 
X, y , 2 
x’, y\ z' 
x', y, z' 
p, q, r 
Poi ^?05 
-j- 
X, y , z 
x', y, z' 
P , , r 
p , q, r 
Po5 '^0 
Po, qo, n 
and the expression for the distance of the two lines (i, yj, 1), (|', ??', f) is 
COS" 
{p^ + qy)+ r^) ( Ppg' + go*;' + + jp^' + gr)' + rt;') (Pog + go>; + rpO 
^^{p^ + qn + rK) [p^ + S'o’J + ^o?)l C 2 ( + r?') + ^o?') 1 ’ 
or what is the same thing. 
sin“‘ 
[qrQ—rq^) (>i^' — + {rp^ —pr^ (g>i' - + ( jogp— gPp) g'»i) 
\^^{pi-\-qy\ + r^) (po? + qon + ro?)] C2(;?p4-<?>)' + r?')(;^o?' + 5'oV + ?o?') ’ 
isr 2 
