86 
ME. A. CAYLEY’S SIXTH iVLEMOIE UPOX QrAXTICS. 
axis being I'), is 
(91, ..II, n, ^)'(91, .-I^'. ^')"sm^^-{(a, ..II, ??, 
( 91 , ..II, n, --II', I'y cos^ O-Kia, hi-l 
from which it follows that the distance of the lines (|, n, 1) and (|', n, ‘C) is 
(^, ..B, >1, V, ?') 
1 ^ . - — . , — n 
s/S. ..I?, ,,W(a. ■ • ■»'. -i'. 
or what is the same thing, 
;/(a?.^iir^\/(^, ..ir,"^^ 
218. And we may from the first formula of either set, deduce for the distance of the 
point {x, y, z) and the line (f', n, the expression 
. >/K(r^ + Vy + ^'^) 
{a , . .B, y, zf \/ (^, . .B', 
as may be easily seen by writing • • toi’ ^ly ■> ~'> oi • • toi 
I, n, and putting sm“* for cos~b 
219. It may be noticed that there are certain lines, Az. the tangents of the Absolute.^ 
in regard to which, considered as a space of one dimension, the Absolute is a pair of 
coincident points ; and in like manner certain points, idz. the inemits of the Absolute.^ 
in regard to which, considered as a space of one dimension, the Absolute is a paii of 
coincident lines. 
220. We may, in particular, suppose that the Absolute, instead of bemg a piopei conic, 
is a pair of points. The line through the two points may be called the Absolute Ime ; 
such line is to be considered as a pair of coincident lines. Any pomt whatever deter- 
mines with the Absolute, two lines, viz. the lines joining the point mth the Wo points 
of the Absolute ; this line-pair is the Absolute for the point considered as a space of one 
dimension or locus in quo of a pencil of lines, and the theory of the distances of line;, 
through a point is therefore precisely the same as in the general case.^ But any line 
whatever determines with the Absolute (meets the Absolute line in) a pair of comadent 
points, which pair of coincident points is the Absolute in regard to such Ime considered 
as a space of one dimension or locus in quo of a range of points, and the theory of the 
distance of points on a line is therefore the theory before explained for this special case. 
But we cannot, in the same way as before, compare the distances of pouits upon different 
lines, since we have not in the present case the quadrant as a unit of distance. The com- 
parison must be made by means of the circle, viz. in the present case an)' conic passing 
through the two points of the Absolute is termed a circle, and the point of intersection ol 
the tangents to the circle at the two points of the Absolute (or what is the same thing, the 
pole of the Absolute line in respect to the circle) is the centre of the circle. The Abso- 
lute line itself may, if it is necessary to do so, be considered as the axis of the circle. It 
